I used to have an '89 Chrysler New Yorker that did that, albeit it did not smell like gasoline as far as I can recall. Turned out that the fuel pump wires had a short somewhere at the pump connector and wiggling the connector would usually start the car... until one day it died completely and needed a new set of pump wires from the dealer. It was one of the MANY electronic / electrical problems that car had though. When the tranny finally went on it and it was sent to the junk yard, hardly anything electronic besides the radio worked on it anymore (and it was only like 10 years old). What a POS that FWD abomination Chrysler called a "luxury car" was, lol!
Back on topic, there should be an emergency fuel pump cutoff switch in your glove box somewhere. Maybe you should check that first and make sure it's working properly? Perhaps it's melted and stuck in the open position due to a short or something... option 2 is a bad fuel pump or fuel pump connector/wires, I'd guess.