Sounds more like a fuel delivery issue to me. You don't mention where you are located but before I did anything else I would attempt to find a known working carb from another TW and install it on yours. That would easily rule out the numerous other issues that could be possibilities.
You should also check the air box for critter nests. Under the seat and behind the battery is the air intake. If it is clogged then no go! Numerous other issues could be causing your problems but you have to start one thing at a time with things like you describe.