Just a couple of generic hints. 1) NEVER use wire nuts on a vehicle, as you suspect they will eventually loosen. 2) Instead, get the proper size but splice connectors, and the proper pliers to crimp the splices. 3) the "D" in LED stands for "diode" that means that unlike incandescent bulbs, LED's are polarity sensitive. If connected backwards they will not light. My guess is that the white wire on your new LED is the ground and the yellow is the power.
Central New York; 2016 Rockwood 2604WS travel trailer
2014 Ram 1500 Laramie Eco Diesel; 2002 Harley Heritage Classic