Water inside axles - normal?

Are you the original owner? Has the trailer been in high water/flood?

drilling a hole will not destroy the axle, even a 1/4 hole or bigger.
Be sure that the water is getting in from rain and not from that time when the axles were completely submerged under water. In either case you do need to get the water out as it is rusting from the inside. A drain hole is nothing compared to rust.
Running brake wires on inside or outside doesnt matter. They are protected from stuff on the inside but obviously could be sitting in water. New wires, no matter composition, should be water proof enough to work.
BTW, brakes are not grounded so you need positive and negative wires run to bakes, never ground the negative to the frame, and there is no positive or negative on the brake, either direction works - it is just a little magnet you are powering up.

I’m not the original owner, so I’m not sure if the axles were submerged. I’ll drill some holes and rewire the brakes in the hopes that’ll fix the wiring fault.

In addition if you are going to run the wires on the exterior of the axles, let them dry out for a while and then shoot some quality rust inhibitor such as Corrosion Free into the holes and then plug the top ones with a good plop of sealer. That will prevent any further rusting.

Thanks for all the great advice. I ordered some jacketed duplex wire to replace the existing wires that cross through the axles. I’m going to replace the connections at each wheel with the 3M Scotchlok 314 connectors. Are these a good option?

Careful with the 3M crimps. make sure the small and larger wires are not next to one another. Separation will give better connection.