Fuel pump should only run for 2 seconds with key on
Once engine is started, above 400 RPMs, it will be on full time
But fuel pump on full time(key on) shouldn't cause a running issue, so there are two issues here
All fuel injection computers have a "Clear Flooded engine" mode
Key on
Press gas pedal down to the floor and hold it down all the way
Crank engine
It should NOT start or fire, fuel injectors are OFF
This tests for leaking injectors
If computer "sees" WOT(wide open throttle) and 0 RPMs at the same time it will shut OFF injectors, so "Clear Flooded Engine" mode
Rich running and fuel pump on all the time can be a computer issue, pull out the computer and open it up
Have a look at the circuit board for corrosion, or "rust"
There will be 2 or 3 blue or black capacitors, these often fail/leak after 20+ years, but under $5 to fix
If you can solder or have a friend that can its very easy to replace these