Coming back to this one, I finally got a day to sit down and try out some troubleshooting.

I put some gas directly into the carb and after a few tries it started up and stayed on. Sounded just fine too. Saw another tip elsewhere to loosen the fuel line going into the carb to see how much gas spills out, and it was just sort of piddling out as opposed to spraying everywhere. I guess it's possible that the fuel line is partially clogged seeing as how some fuel was getting through, but it seems more likely it's the pump.

Gonna try and track down one of those now. Thanks once again everyone.