Hello, and welcome to the forums!
It sounds like you might have gotten a china knockoff carb online, rather than a much more expensive (obsolete) OEM Honda carb. If so, you'll need to start over... china carbs and parts are useless garbage.
The best way to fix your TRX350 is buy an NOS Honda or a Shindy carb kit and clean and repair the original Keihin carb.
Keep us in the loop... and if you have some pics of the old gal that you'd be willing to share... we'd love to ogle them!
BTW, you can get a free copy (PDF) of the factory service manual for your old tank
HERE . It is an invaluable source... print it out for best results.