My '04 Grand Prix with 130k miles has a few issues that may or may not be related, so I will list what seem to maybe be connected.

The vehicle has for as long as I can remember have difficulty starting on a hot start without giving it gas while starting the car. About a month ago my Check Gas Cap earning came on, followed soon after by an engine light. Neither has gone away by changing the gas cap with a new one. About the same time the hot start issue has gotten much worse, with the car dying when turning the ignition unless I give it quite a lot of gas. On top of those, and about the same time as well, I'm getting a weird issue where I cannot fill up my gas tank fully. The pump cuts off like it is full after just a gallon or two make it in. Sometimes I have to keep waiting a few seconds then pumping more and it cuts off again. Sometimes I can just let the gas flow slowly until full. Almost like the gas isn't get into the tank fast enough and backing up. Not sure what that's about. 

Now, in the past couple of days I am getting sputtering while accelerating. I planned to take it in for a tune-up which is certainly overdue, but if this may be more clearly connected to the gas cap and hot start issue, I'd love to hear any ideas.