Backup Lights Not Working

The switch has 3 terminals. Reverse lights the switch jumps the two outer terminals of the switch. So remove the switch connector and with key in "run" see if you have power to one of the outer terminals. If you have a helper, jumper a short wire across the two outer terminals and see if lights work.

As suggested, otherwise check bulbs, and with key "on" and in reverse, check for voltage at bulb terminals.

You have a shop manual? Download one free from MyMopar. No I don't work there
yep, I didn't want to send him back under the car or under the dash right away. Some testing can be done under the hood, at bulkhead connection. I have pulled the connector off the 3 pin NSS switch and used a cotter pin to jumper the 2 outer contacts.
I recall one case where everything was good except sifter linkage adjustment. The trans was in reverse but adjustment was off just enough to open the switch. Reverse lamps blinked on and off again as shifter moved. A helper came in handy there.
Neutral is a wider detent so we can wiggle the shifter there and get neutral safely to work. Park and reverse are a lot tighter and need to be on the money. Worn bushings in shifter linkage, failed/sagged trans support, etc.. can adversely effect this adjustment. Lowering the steering column for some other issue in column shift builds is even more common loss of linkage adjustment.
He didn't say it wont start in park but does start in neutral so maybe linkage adjustment is not the problem. New switch doesn't mean good switch today.
There was another thread here where the owner had a replacement switch that was just a little bit longer than the one he removed. It worked only once and failed. Took it out again to find it broken. Good luck to all.