Just a guess, that's usually my problem when I can't keep something running, though I've never had anything that old, so it could be more complicated. It's usually best to start with the simple things when troubleshooting. I spent a week and took a bike halfway apart to find that I'd unknowingly bumped the kill switch once.
The carburetor jets, especially the idle jet, are very small. Cleaning them after they've become restricted with old fuel (smells and acts like furniture varnish) is generally not going to work.
You need to get a carburetor rebuild kit, it'll have all new jets, float bowl gasket, float valve and more.
The TRX300 kit is still readily available and is probably in stock at Friendly in Baton Rouge.
One thing short of a carb overhaul you can try is to let it run on a concentrated fuel and Seafoam mix. Seafoam is available at most auto parts houses and even most Walmarts. It will clean those small passageways that you can't. It will clean carbon out of the cylinder and off the piston and valves. It will make your neighbors think you set your house on fire as the exhaust will smoke like crazy. You will need to replace your spark plug after using a concentrated Seafoam treatment.
Check the slow jet. When a bike won't idle, even warmed up, that is likely the problem. I seem to have had to clean or replace mine every couple of years when I had ATV's, despite gas treatment and draining the carb.
I've been a motorcycle tech for 20+ years.
As others have said, sounds like pilot jet (aka slow jet) is plugged.
Either varnish or mildew from ethenol.
Clean or replace jet, clean other gunked up items and fresh fuel, give it a try.
We try to run non-ethonol fuel with heavy mix of marine formula Stabil in all our small engine and seldom have to clean a Carb.
Good luck, hope that's all it is.
No one will be able to definitively identify your issue off your original post, so it is all a guess. Sounds like your running lean due to restriction in the idle circuit, most likely from bad fuel (sat to long without stabilizer). as suggested, cleaning the idle jet may work. Rather than replacing jets, I would completely disassemble the carburetor(s), setting aside anything made of plastic or rubber or paper. Any adjustment screw you remove, be sure to bottom it out first, writing down how many turns it took, so that it can be adjusted to that position again. Soak all parts in B-12 chem dip http://www.berrymanproducts.com/pro...rryman-chem-dip-carburetor-and-parts-cleaner/ available at all auto parts stores for a couple days. Reassemble, positioning idle mixture and any balancing screws you removed to the same position. It should run great. Many times the varnish restricting the gas flow is in the small passages in the carburetor body, and hosing it with cleaner just won't get it like a good soak in the cleaner bucket. Use fuel stabilizer (Sta-bil) in anything you let gas sit in for a month or more (anything but your daily driver.) Just a personal observation, gas doesn't seem to last as long as it did 10 years ago or so. I figure the gas company's used to put some form of an octane stabilizer in the gas mixture and have stopped to keep profits high (just a wild guess, It just seems to not last as long as it used to) making stabilizer more important than before.
Had this happen dozens of times with various bike and always either trash or water in the carb. Just the condensation can form a small drop of water in the carb and that single drop can restrict a jet.
Simple solution is to be able to remove and disassemble a carb and put it back on. Don't wast money on a carb rebuild Killy unless it has sat for a long time and crusted over. Be careful with the bowl gasket because once it is removed and especially if hit with carb cleaner, it swells and will not go back in place. There are simple fixes for that too though.
Take the money someone suggested spending on a carb kit (around $30) and buy a set of torch too cleaners, a couple cans of carb cleaner and replace the Phillips head bowl screws with Allen head screws, put the other $15 back in your pocket.
Disassembling and reassembling this carb is dirt simple and self explanatory, just put everything back they way it was.
this is for EVERYONE [ except general mills and Cochise ].
I have Nothing against Seafoam, HOWEVER , for some reason people think seafoam is the BOMB.
Berrymans Chemtool is 100% better.......... Why you ask, Because it contains a chemical 2-Butoxyethanol
Take Seafoam then add 2-Butoxyethanol and you end up with Berrymans chemtool.
The concentration of 2-Butoxyethanol in Berrymans is a little weak. [ about 5% ]
[ It has to be for commercial over the counter sales, if used improperly you can do some serious damage ]
We use a 30% mix and let me tell you , you would be amazed at what this stuff will dissolve .
So pass up the Seafoam, and start using Berrymans [ follow mix instructions on the can ] ...... Save ~$3 and be amazed at how much better it cleans.