I have a 1997 Grand Am with the 3.1L. It recently has been giving problems staying running when cool and it hasn't been driven for a 2+ hours. It turns over fine, and fires fine, but all the dash lights stay on, the gauges don't work and it will stall out after a second or two. If I press the gas and rev the engine it will eventually stay running as long as it's kept above 2000 RPM. Eventually at 2k+ RPM the gauges will start working and all the dash lights will go out like normal except the "Theft Sys" light. It will run and drive fine after that unless there is a large amount of sitting and idling, then the cycle above starts again. If driven for 20+ minutes and restarted within about an hour it starts and runs fine. It seems to have less issues the warmer it is out as well.

I replaced the battery thinking maybe it wasn't holding a charge, but it didn't help. Also tried disconnecting the battery when not using it to ensure there was no power leech which had no effect.