My radio fuse went out. When i changed it my radio will only stay on when the car is not turned on but when i start the car the radio just shuts off. What are some other possible problems?

internal circuit fail

Sounds like an electrical issue... When was the last time you changed your batt.? It could be that you need to change your battery.

check fuses and connections

Your accelerator pedal is bad. The IVS (Idle Validation Switch) is probably not shutting off, or the potentiometer in the pedal went bad.
Hopefully you do not have adjustable pedals as the pedal assy. can be expensive for those.

it may have been wired direct to the fuse power! you may have to turn it off manually! or change wire to a fuse that shut off when you turn off key!

GET A GOOD FUSE TESTER, TEST EACH FUSE. THERE ARE 2 FUSEBOXES. ONE UNDER THE HOOD & ONE IN THE DASH. ALL FUSES CAN BE TESTED WHILE STILL IN PLACE. THERE ARE TEST POINTS FACING OUT. TURN ON IGNITION SWITCH FOR THIS TEST. POWER ON 1 SIDE, BAD FUSE POWER ON BOTH SIDES, GOOD FUSE. POWER ON NEITHER SIDE, SPARE FUSE.

Check and make sure the ignition switch is not worn out and allowing you to remove the key when the switch is turned backwards in teh accessory position. once the key is removed and the radio is staying on, turn the ignition without the key in it, both directions and see if it shuts off, if it does, menas work cylinder or key.

You have 2 fuses/power wires for that, 1 will be constant power and other will be when key turned on.

I have the SAME EXACT problem. I thought it was the radio - bought another - went to change the units - heard an electrical arc - now the car cranks but doesn't start and the new radio doesn't work either!!! (not that it matters anyway)
Is there an electrical problem with '96 Lincoln's. Where should I start to diagnose the non-starting issue?

Sounds like the fuel pump is shorted.