Q: Starts up and dies down after driving through rain

Driving in the rain. Drove through a puddle (not deep but enough to splash up water) oil light and battery light came on about 5 minutes later while driving and the car died. It'll start after sitting for about 20minutes but then it chugs and dies. I try to give it gas but it doesn't rev up just keeps chugging until it won't start for another 20mins. It runs for maybe 1-5 minutes and cuts off. Electronics still works. I think it's the fuel pump because I cannot hear it come on.

My car has 209417 miles.
My car has an automatic transmission.