I would correct this. That's almost certainly exacerbating the problem as it sounds like it's still pumping fuel into the engine when it's not running. There is zero good to running the pump longer than the ECU wants it, and plenty of risks.
I would recommend wiring up the self-shutoff control like it should be (that's what one of those diodes is for).
Jumper the read mode connectors (or, ground the wire from the ECU if those connectors are gone), and the CEL will flash out codes.
Yea, figure out if it's just flooding one cylinder. Check the vacuum line after the FPR for fuel. Grab a noid light to see if the ECU is holding that injector open.