Got the 2 lights replaced this morning.
While the washers were being put into place, I decided to use the old lock washers, as well as the new washers. Light - bracket - flatwasher - old lock washer - new lock washer - new nut.
The wire on the trailer side of the splice had a small cut in it. I used a pair of wire cutters and cut the wire where it was exposed. The wires were twisted together, a cap put over the splice, and everything wrapped in electrical tape.
To keep the wires from flopping around with the trailer is in motion, I used a small zip-tie to secure the wire to a hole in the light bracket.
Once everything was in place, the lights were tested, and everything worked.
Now I dont have to worry about getting a ticket for my trailer lights not working.