Sounds like what my Burb was doing this time last year. I did not want to believe it was the fuel pump either (since you have to drop the tank and I was NOT looking forward to that!). Checked everything else, moved the fuel pump relay around, etc. and guess what it ended up being?!
That's right, the fuel pump! My neighbor and I spent our Labor Day off replacing it; fortunately he had done one on his brother in law's GM SUV and so he had some idea of what we were getting into.
I checked the test port on the fuel line that
@someotherguy is referring to. Very little sprayed when I pushed the pin on the Schrader valve (like what a tire valve stem has inside).
How you check this without a gauge, is to unscrew the cap, (hold onto the cap or put it somewhere it won't get lost) and push that pin down with a small screwdriver. Have a clean rag or shop towel around it to catch any spray; you DO NOT WANT any of that gasoline or vapor to get into the ignition and this is not very far from the distributor!! If you get a strong stream, you'll need to get a fuel pressure gauge and test the fuel pressure by attaching it to this test port. If none or very little comes out, the fuel pump is probably bad. At least that's my experience!
The new pump will probably come with a new pigtail for the connector plug; GM made a running change on these to a different design and shape.
