Why is it burning? LED lights do not require a lot of power.. you "Broke" the white (Ground) wire but the white wire and the neutral form sort of a capacitor (NOTE the black and ground do as well).
What is happening is the load from the LED's is not high enough to "overload" the capacitor formed by all that romex between switch and light, Thus the light is seeing likely 60 volts or close to it. Possibly less.
The solution is simple.. Switch the proper wire.
you may stop reading here.
One of the issues is that We RVers live in a dual world AC and DC.. Many things work 100% on DC, and may even work will on A/C if the current is high enough.
But A/C is a different animal.. As an ELECTRONICS technician I know this, but .. well, that's the difference between Electronics and Electrical.. The parallel wires, at DC, form an open circuit
But at 60Hz it's a high value resistor.