Found a few bits of info that may help. It looks like maybe the two switches have traded places at some time back? The drawings show those two reversed but no big thing! Click these snips to get a better look at them.
On this drawing on sheet 4 Ifound the switches and they give some info on wire ID used. Winnebago stamps ID on each wire, if they are still good to read?
Like this example.
If you can read the ID on the wire, then go to this list to "decode" where the wire comes from and goes to as a way to tell what it does.
Given that info, the list tells us there should be a wire labeled KF that is ground. So a check with a meter to make sure that ground is good would be one good step to make as that wire likely goes down underneath the RV to a big copper bus bar bolted to the frame. Many are right behind the batteries but that may be different but almost always good and solid on the frame. The problem is that area is where it gets lots of water and dirt which make them suspect if we are not getting good ground to something!
Wires LG and LJ both go to the relay, one to engage and one to disengage the relay! But the one labeled LK may be confusing you as it shows it works the light and it may getpower from a totally different place, so that it lights, even if the switch is totally no good!
One quick way to test is to hook a wire to some metal part that is ground or the negative post of the battery and then hold it on the pin for the ground wire at the switch whil you push the switch each way. In other words, wire around any bad ground wire to see if that makes it work!
This switch sends the signal to the relay which is often outside in a compartment. Maybe behind the drivers side? Behind a label like this?
If you take the breaker/fuse panel out of the way, you find the relay on the right side as this drawing shows and that is where the wires from the switch wind up. When working, you can feel or hear it click as it moves, if close.