Sounds like you have a mystery corrosion issue. Like suggested, measure the voltage input to the string...I think you'll see 3-4V there. Even with the switches Off, unless you actually disconnect the battery grounds, there are several 'live' circuits in a typical Class A. One of those live wires is probably supplying voltage to the light string though corrosion or a single strand short.
For example, in my rig, even with the salesman's switches Off, my ceiling fan will run. I have no idea why it's wired that way. Probably an afterthought or a mistake at the factory.