Have you drained the float bowl?
If not, remove the fuel solenoid to empty the bowl (if no other bowl drain is present). Catch the contents of the bowl in a clean, clear container that will permit you to examine what comes out of the bowl. If you see slome slugs of water below the gasoline, that is likely the root of your problem.
If you see debris that looks like solid particles, you may be seeing stuff that has sloughed off the inside of the fuel hoses between the filter and the carb.

Thanks for the suggestion.

I removed the fuel solenoid and saw one/two small white flecks which I assume came from the bowl.

I sprayed out the bowl and external linkages with carb cleaner. Then re-installed the solenoid.

I finished up with spraying down all of the cables / linkages with WD40.

The engine now dies every time I engage the deck, but otherwise, runs great.

I know pulling solenoid didn't do this, so thanks again for the suggestion.

Before this, I had mowed for about an hour with only a few hesitations / stutters from the engine. But like I said, now she runs great until I engage the deck, then she dogs down.

Anymore thoughts?

OK, please post the "Technical model number" of the tractor. This is found on the Sears data decal, usually find this decal under seat or hood. With that, maybe we can view a wiring diagram just to make sure no electrical switch is at fault here.
In the meantime, does it die only when you engage the deck? or will it also die if you try to accelerate in gear from a stand still (like "drag racing").
I ask because both actions apply a sudden and substantial load to the engine.
If the engine bogs and stumbles when you try to accelerate the tractor suddenly, that pretty much shouts "insufficient fuel delivery" by the carburetor.

Usually, it is necessary to remove the carb for disassembly and cleaning. This means removing the jets from the carb to clean them and their passages with fine wire and spray carb cleaner, accompanied by blowing out all passages with compressed air.

But we need to check all possibilities.

Thanks again for responding!!

Model# 917.289900

I tried the drag racing scenario and it sets me back in the seat with no hesitation. Tried the deck and it immediately dogged down.

Sears believes it is more important to have half the owner's manual en Espanol than it is to feature a wiring diagram. so the online manual is no help in that area.
But, we can try some things before we start buying parts.
Can you remove the deck drive belt from the PTO clutch? Tie the belt away from the PTO so it can't touch any moving parts.
Start the engine and run it at the RPM you have been using prior. Engage the the PTO switch. If the engine dies now, move the key switch to the ROS position of the key options. If the engine now runs when you engage the PTO, your reverse indicator switch is defective or it make have been knocked askew somehow. Find the "reverse gear indicator" switch on the transaxle, or it may be on the shifter linkage somewhere, look for it to be associated with the linkage that moves when thr reverse pedal is activated. Post back your findings on removing the main deck belt and/or the ROS position results.

Alright, you're close to being my hero!!

I swore I tried this a couple of weeks ago, but...

I removed the mower belt; started the tractor; and moved the throttle to max. When I engaged the PTO for the deck the tractor started to die. As soon as I disengaged the clutch the engine sprang to life. I did this several times with the same result each time.

OK, now, unplug the wire pigtail that connects to the clutch under the tractor, tie the wire where it can't encounter the moving crankshaft and PTO.
Repeat the test as before. If the engine continues to run with the clutch disconnected, we will need to test the clutch for an electrical short.
Did you test the "Reverse Operation System" switch option on the key switch as I suggested?

I had the same results with the key switch in ROS. I forgot to post this the first time.

I disconnected the clutch connector (2 green wires); started the tractor in high speed; engaged the clutch; and the tractor started to die. Disengaged the mower clutch and all was good again.

Quick look at the schematic and I'm thinking something with the seat switch.

You have a schematic? Can you post an image to the thread or send me an image in an e-mail?

Sorry, I don't have a way to scan it in. I was trying to find an electronic file to send, but was unsuccessful.
I checked the seat switch and it's contacts reposition correctly, based on continuity changes.

I believe my problem is a pinched wire bundle. There are 3 wires coming from the seat switch connector which run up to the engine area to a 4 wire connector. With the 3 wire and the 4 wire connectors disconnected, I read continuity between all three of the wires going to the seat switch. I noticed the wire loom runs from the seat switch towards the engine on the outside of the frame rail. Except about half way through the run the loom disappears between the rail and the body and then re-appears about 1 1/2" later. So I'm pretty sure the bundle is pinched and finally shorted.

Sound reasonable?

For the moment......forget about pinched wires and such.
Pinched wires and chafed wires are about the least likely things to cause electrical problems, especially if the wires are bundled into a harness. It takes some kind of physical contact with metal surfaces to pinch or chafe wires and you should be able to tell whether that condition is present (the point where the rub or pinch occurs) by a good visual scan.
Not having the benefit of being able to see a wiring diagram for your tractor adds to the problem of helping from afar.
So, let me ask you about some components that may, or not, be shown on the schematic you have. Post whether the following do, or do not, appear on your schematic.
(1) Operator's Presence Relay. There may be more than 1. If there are more than 1 OP relay, tell me how many you see.
(2) Reverse Switch
(3) Seat Switch with the words "shorting connector". The phrase "shorting connector" is an important detail in knowing the wiring configuration and how it will affect testing of circuits.

Answer those questions and then e-mail me. I will pick a Craftsman diagram and send it to you so you can tell me if it matches your schematic.
If we can work off the same sheet of music it will be better in determining what you need to check for.

The wire bundle was severely pinched between the body and the frame. So, I cut out the damaged wires and soldered in splices.

I think it was too late last night as I thought the problem was shorted wires. Unfortunately, I didn't understand where the shorting switch was (appears to be part of the connector to the seat switch). So my "shorted" indications appear to have been the shorting switch. I started the tractor this AM and I am able to engage the clutch without stalling the tractor. I'm headed out to put the deck back on so everything is back to normal. Then I'll run it again.

In the mean time, heres my schematic:

http://i762.photobucket.com/albums/xx268/08fordsd350/scan0001.jpg

Here is a link that might be useful: {{gwi:348487}}

Thanks for the schematic, you done good!
I'll save a copy of this for future reference.
At least we now have the same "picture" to look at if further tests are needed.

Thanks for all of your help. It appears the problem is corrected - though only time will tell when it comes to sporatic problems.

The wire bundle was pinched so tight between the frame and a bolt head that I had to cut it apart to get it out. It appeared that two of the wires had punctures through the insulation which I assume was from the bolt head. So I can see how various weight shifts during mowing could add / subtract from the pinch causing the intermittent problem.

Anyway, thanks again. Wished I'd have thought of my wife's scanner last night, so I could have gotten a schematic to your sooner.

By the way, does the switch development for the ignition switch look correct to you? If I'm understanding it correctly, the only time the M terminal is in the circuit is with the switch in OFF. That doesn't seem valid. Guess I should check it against some actual voltage measurements.

Thanks Again!!

***"the only time the M terminal is in the circuit is with the switch in OFF"***
Yes, that is valid and correct.
M is for "Magneto". This engine, as most OPE, uses a magneto based spark ignition. Magneto coils produce their own electrical power for the spark when a magnet on the flywheel passes the magneto as the crankshaft spins.
Magnetos do not need any external battery power to operate.
What you see on the schematic is how the engine is "killed" when the key is turned OFF. Turning the key to OFF connects the magneto kill circuit to ground, and this stops the magneto from producing spark. Most seat switches and interlock systems also have a way to connect the magneto kill circuit to ground to kill the engine the same as if you turned the key off. I suggest you be very careful when checking circuits wherein you might be "probing wires" with a "hot wire"!!!!!!! If you inadvertently send 12 volt power back on the magneto kill wire, it will do exactly that.......kill the magneto! Then you can post and tell us what a Kohler magneto costs :^)

Oh, I guess I spoke too soon when I warned that chafed or pinched wires seldom turn out to be the culprit. Just my luck to say that and then have your issue be caused by that.
Realistically speaking though, chafes and pinches are rarely at fault, so it's usually best to check the things that can be checked without any hacking first.

Glad you followed your intuition based on the evidence you saw and are now up & running.

The schematic IS on the Sears website, along with all the other parts diagrams-

Thanks Bill, I hope my red face doesn't show too bad.
It's good to know that I can still find schematics if half the Owner's Manual is in spanish instead of having a schematic. Never had this pointed out to me before now.
I always went straight for the Owner's Manual previously.

Mownie-
That's the first time I've seen one like that with the parts diagrams, so I wouldn't be too red faced!

Thanks for the reply on the magneto kill ckt. operation.

The tractor is running strong with no deck issues so far, so it looks like a success.

Thanks for all of the help.