That sounds like a classic clogged fuel filter.
Get it up on jackstands or on a lift at a local mechanic and follow the fuel line from the tank (up front) back to the engine. The fuel filter could be anywhere along the length. Once found, replace it with one similar. While checking, look for any sharp corners in the line that might be crimping it partially shut.
If it is a see-through plastic filter, please do not use another one of those. Instead, go to a local NAPA store and ask for a NAPA Gold 3031 metal-can fuel filter and use that instead. Much better filtering and it won't melt under heat and set your car on fire.
You probably won't have to pull the tank (but it is really easy to do, no matter what). Instead, you can get to the outlet under the tank by pulling the passenger side front wheel to gain access to the underside of the tank. Drain the tank and then remove the outlet (there will be some gas spilled even if you drained the tank) and check for debris or clogging. There may be a small screen on the outlet up inside the tank - Or maybe not. Use teflon tape on the threads when you replace it.
The only other things affecting it might be (a.) a rubber fuel line that has grown mostly shut, (b.) a clog at the carb pilot valve (unlikely, given that it runs for 15 minutes) or (c.) a weak fuel pump (again, unlikely).