More than likely, your main jet has a build-up of varnish. Since the unit is running, you might be able to clean it up with a tank of gas and cleaner, but the best way is to remove the bowl nut on the bottom. I don't know if that is the jet, or if it's up inside.

These carbs can be difficult to access if there is no cutout.

Being non adjustable doesn't cause this problem unless the governor spring is messed up. If it bogs down under a load, but runs at normal speed when started, it's almost certainly the carb.

Look at this thread for examples of main jets.

http://forums2.gardenweb.com/forums/load/lmower/msg1115385424964.html?8

Here's a link to the service manual for this engine.
Since it's a 2 stroke, you probably don't have a "conventional" carb/bowl nut jet, but I haven't looked.

Also note-
"EXHAUST PORT CLEANING. The exhaust ports should
be cleaned after each seventy-five (75) to one hundred
(100) hours of use."

Here is a link that might be useful: Manual

I checked and all the passages in the carb are clean. It does have a float carb with the bowl nut as a jet. I was thinking about maybe drilling out the jet since it seems to run super lean

You will probably have to remove the carb at this point and clean the internal passageways. In particular, the passageway that that is inside the center tube that the jet screws into. You probably have a Welch plug in the carb and if you don't see full flow fluid coming out of that tube passageway, the engine won't work right. I've seen it a thousand times.

Your carb is probably very similar to this carb. Look at this former thread. In picture 12, you will see the red plastic tube inside the center tube of the carb. At the 12 o'clock position above the red tube is an opening. If you squirt carb cleaner in that hole, it must come out somewhere else. In the example of the carb shown, it will come out at one of the holes beneath the Welch plug in picture 13. There are no exceptions.

http://forums2.gardenweb.com/forums/load/lmower/msg061726346162.html

Since you are able to get this engine started, and keep it running, there is a chance that you can clean out the carburetor by adding Gum-Out, or something similar, to the gas, and then running the engine for at least 30 minutes. This might work, however, you will need to be aware that solvent type cleaners can damage plastic parts, and also rubber fuel lines, so it is important to use up any gasoline that has been treated with Gum-Out, and then fill the tank with fresh gas, untreated.

And definitely DO NOT clean the exhaust ports because that OBVIOUSLY could NEVER be the problem.

Bill, you are SO wrong about that. :)

If it starts normally but won't rev up to full RPM, I would definitely check the exhaust port or spark arrestor on a smaller tool.

Since the OP ignored the recommended suggestion to clean them in my first post, I thought he might ignore me twice.

Well if you want trial and error , then use the fuel cleaner route 1st if this fails then inspect and clean the exhaust ports , finally if no improvement disassemble the carb and use Bays tried and proven internal cleaning of the tec's problematic carb internals. A 5 yr old 2-stroke engine could have issues within carbon fouling of the exhaust if cheap oil or over rich ratio was used . Normally the unit wil run extremely hot with reduction of power . This again is evident within a lean running 2-stroke also lol . Hope the Op gets back to all with follow up info !