Ah yup, you have the old Kelsey Hayes brakes that are now out of business.
The good news, your brake shoes have a good amount of lining still left on them. Since they where not working, they can't wear out the lining.....So you do not need new shoes at this point unless under close inspection you see a problem as getting parts for these old brakes is really hard.
You can take the brakes apart and clean them up so everything moves like it should. Also these old Hayes brakes have the adjuster up on the top which makes this really hard to adjust once the drum is on. In fact, I do not know if you can adjust it once the drum is on. I have never did one of these older ones.
That said, you need to tweak the adjuster and play test fit with the drum so they are close enough before you put the drum on. And to try and get the slight drag the same between the left and right wheel.
Yes, the wiring can be bad, especially inside the axle tube. It is a classic camper brake failure when the wire inside the axle tube gets old and brittle. And as the camper bounces down the road, the insulation gets rubbed and creates a short and you have no brakes then.
Since you have a small camper and only 1 axle, suggest you run a new hot wire and a new ground wire down each side of the camper to each wheel. One set for each wheel. Then you have no wires inside the axle tube. When you make the wires jump from the frame to the brake coil, leave enough of a loop so when the axle moves up and down you do not stretch out the wire and break it. And fasten them to the frame so the wires do not get caught up in the wheel. They sell small bolt on one hole nylon eyes at the auto stores that work well for this. They look like this. Just get them small enough for your 2 wires they will grab the wire
For your camper, I would suggest to use no 12 awg copper stranded wire. This is heavier then needed but gains you a benefit. By going direct to each wheel from the front it will lower the voltage drop with the heavier wire and you eliminate the axles wire tube issues. You will join the 2 brake hot wires up front and then to the brake wire on the 7 wire cable to the truck. Also join the 2 ground wires and attach that to the ground wire of the 7 wire truck cable too. Do not use a trailer frame ground as corrosion on the ground can be a bad actor in a brake circuit.
Make sure the connections are good and tight and completely sealed. Do "not"
use scotch lock quick connectors. These things. I see these all the time on trailers and they are a disaster waiting to fail.
Ideally you solider the connections, shrink wrap and then 3M 33 tape them to keep the water out. If this is too much for you, then a good sound butt splice that is then tapped to keep the water out will work. Water in the connections is a bad thing.
As long as the brake coils are still good, which they very well could still be, you should be/may be able to clean up the old brakes and get them working again.
Let us know how you make out.