Sputter, loss of power and backfire

1994 Deville 4.9 159000 miles. I originally thought I got a bad tank of gas, because I started having this problem right after a recent fill up. The car will sporadically lose power, sputter like it is either starved for gas or getting too much gas, and will backfire. If I turn the car off for a few seconds and start it back up, the problem seems to go away. It will drive fine for awhile. Sometimes a day, but then the issue comes back. I have tried fuel cleaners and fuel treatment in the event it was bad gas but that has not resolved the issue. I checked the plug wires and all appeared to be good and connections seem to be okay.

On a side note, when I turn the key without starting the car, afer a delay of about 5 seconds, I hear a motor running under the car that doesn't stop until I turn the ignition switch off. I thought it may be the fuel pump but it doesn't sound like it is coming from the tank, but the back end of the car instead.