Backup lights won't work properly

Backup Lamps
The B/U LMP fuse #42 in the underhood junction block supplies ignition positive voltage to the park/neutral position switch. When the driver places the gear selector lever in the REVERSE position, the current flow is from the park/neutral position switch to the backup lamps. The backup lamps are grounded at G401 and G402.
Are you also having shifting problems ? If not , I can tell you it's not the fuse ! Wiring problem, the bulbs , park/neutral position switch !
But

It depends,both a DOHC and SOHC version were made. http://www.allpar.com/mopar/2.html

Is the car still in gear? If it is, then it will not start till you put the car in park. I've done this before and the same thing happened, had radio, lights etc...till I put car in gear and it started. Check the gear selector, if it's not in park, then put it in park and try to start. Hope that helps.

shifter linkage needs adjustment or something has fallen in to shifter mechanism

gearbox selector out of adjustment at the gearbox
needs to be lined up in neutral position and rest should work

internal park locking pal worn

an engine/gearbox mount broken or sagging

try taking the center surround out and checking to see there is nothing under the gear selector stopping it from locking in park

Your safety interlock switch is going out or is not adjusted properly. There is a procedure for adjusting the switch. The switch is on the transmission in the engine compartment. If you want to adjust the switch, loosen the botls holding the switch and follow steps 7-8 below. Else, you can just replace the switch entirely by following all steps.

The neutral start and backup switch are one unit mounted on the top left end of the transaxle. The neutral start portion of the switch allows electrical current to travel to the ignition system when the shift selector is in park or neutral only. The vehicle will not start when the selector is in any other gear. The backup portion operates the rear backup lamps when selector is in the reverse gear.

REMOVAL & INSTALLATION See Figures 1 and 2

1. Make sure the shift selector in the Park position, then apply the emergency brake.
2. Disconnect the negative battery cable.
3. Disengage the neutral start switch electrical connector, then remove the shift control lever on top of the switch.
4. Remove the two neutral switch attaching bolts, then remove the switch.

Fig. Fig. 1: Neutral safety switch location-vehicles equipped with automatic transaxle

Fig. Fig. 2: Adjusting the neutral safety switch

To install:

1. Install the switch on the manual shaft.
2. Loosely install the two attaching bolts and washers.
3. Insert a No. 43 drill (0.089 in.) through the hole.
4. Tighten the attaching bolts to 7-9 ft. lbs. (9-12 Nm), then remove the drill.
5. Engage the switch electrical connector, then connect the negative battery cable.

1. put adrain pan under the transaxle.
2. Unscrew the switch from the transaxle case, fluid will pour out.
   
3. Move the shift lever to PARK, then to the NEUTRAL position, and see if the switch lever fingers are centered in the switch opening.

To install:

1. Screw the switch, with a new seal, into the transaxle case and tighten to 24 ft. lbs. And the test the switch with the test lamp.
2. Add the proper fluid to the transaxle to bring it up to the proper level.
3. The back-up light switch circuit is through the 2 outside terminals of the three terminal switch.
4. To test the switch, remove the wiring connector from the switch and test for continuity between the 2 outside pines.
5. Continuity should exist only with the transaxle in the REVERSE position.

Sounds as if you need the electrical portion of the ignition switch. The contacts inside are most likely worn out.

the relay is located on left side frame rail under battery inside of frame