Is this a new problem, or do you not know about the reverse switch?

USING THE REVERSE OPERATION SYSTEM -
1. Depress clutch/brake pedal all the way down and hold.
2. With engine running, turn ignition key counterclockwise to ROS "ON" position.
3. Look down and behind before and while backing up.
4. Move gear shift lever to reverse (R) position and slowly release clutch/brake pedal to start movement.
5. When use of the ROS is no longer needed, turn the ignition key clockwise to engine "ON" position.

Craftsman lawn tractor M#917276390 will not engage in any gear.

Craftsman lawn tractor M#917276390 will not engage in any gear.

You say when it's in reverse, it kills the engine.
Then you say it won't engage in any gears.
How is that possible to be in reverse when it won't engage in any gear?

Are you using the proper technique?
TO MOVE FORWARD AND BACKWARD
The direction and speed of movement is controlled by the gearshift lever.
1. Start tractor with clutch/brake pedal DEPRESSED and gearshift lever in neutral (N) position.
2. Move gearshift lever to desired position.
3. Slowly release clutch/brake pedal to start movement.

You have to be more specific about what does or doesn't happen.

After the engine is started and I attempt to put it in gear it does not engage. I can move the gear shift in all gears one through six and nothing happen. When trying to put it in reverse it kill the engine. I can not explain it any other way.

I'dcheck to make sure the drive belt is turning the pulley to the transaxle and that the transaxle pulley is actually turning the input shaft to the trans axle.

Killing the engine in reverse-
Is the mower deck disengaged?
If not, are you using the correct reverse mowing procedure?

IF yes to both questions, make sure the plug to the seat switch is fully seated and that the seat switch plunger is moving far enough.

I asked if this was a new problem, which you failed to answer.

If you try to move the shifter to REVERSE....are you activating the ROS switch before doing so? Or are you just willy nilly shifting in an attempt to get some kind of response out of the transmission?
It sounds to me like the free wheeling lever (if so equipped) may have gotten moved and thus the transmission is unable to drive.
If no free wheeling feature is on this tractor, some of the shifting linkage may have come apart, leaving only the lever on the fender to move and operate the ROS interlock module, but doing nothing to affect operation of the transmission itself.
The fact that the engine dies when you shift into reverse is probably due to the normal operation of the Reverse Operating System doing what it is designed to do if you aren't activating the ROS switch first.
So let's forget about "dies in reverse" as a symptom of your transmission problem if you aren't using the ROS switch.
What say you?

I would to thank all of you for your help. I have resolved the problem and mower is running like new again.

That's great news Tom!
Would you like to play forum and share with us what resolved the issue? Pretty please.