I got 13.43 by checking the negative of one battery to the positive of the other.
I suppose this means that the trailer is properly wired ?
No, it doesn't mean that the trailer is properly wired but it means that the first item on the trouble shooting list can be checked off.
I'm very suspicious of several outlets having a problem at the same time. As others have mentioned checking the fuse voltage would be a good next step. You can remove the fuse and check the voltage at the fuse socket. Pos. probe to fuse socket, neg. to any ground. Also, on mine at least, the rear side of the 12 volt outlets are accessible. The pos. terminal is exposed. You can also check the voltage there.
There's lots of experience on the forum. Just keep posting your findings and the problem will become clear at some point and you'll have had some trouble-shooting practice.