I have a 1987 GMC 7000 with the eaton 2 speed rear axle. I purchased it an hour away and drove it home no problem. The first trip I made loaded with slag the rear hubs got so hot that the seals melted and oil was dripping out. The hubs and rims were so hot that the oil ignited instantly when it dripped on it. I can haul 8 tons and was only hauling 7-1/2. I changed the bearings on both sides and new seals. I purchased a used service manual for the truck and inside was a service bulletin stapled. It stated on a certain range of axles that a street elbow was added to the fill plug to raise the fluid level in the pumpkin to prevent overheating. Mine fell in the range so I followed the bulletin. I also changed the brakes. It has rear disks and I changed both calipers and the pads. I made 110 hour long round trip hauls and could easily touch the hubs which were barely warm. This year on my first trip to the slag yard I felt the hubs (as I've made a habit of doing) and they were fine. I got loaded up and drove the 1/2 hour home. When I got to my farm the hubs were so hot I couldn't put my hands on them. Has anyone had these trouble or have any suggestions.
My experience has been brakes dragging on these things. I had a '91 topkick that did the same thing. It would never melt the seals out but it would get firecracker hot randomly on its own. A new master cylinder and brake lines fixed mine. I fought it for 2 years like that.
I have an '87 c7000 also, if I let it sit it'll get a sticking caliper that'll make the brakes drag. It usually self resolves.
I just replaced all the steel lines on my C70 and bled the system real good. I had 3 pin hole leaks in the hard lines from rust and that kept the rear calipers from fully releasing. The brakes are 100% better now, the truck will put passengers through the windshield if they get testy. I'd try getting a laser thermometer to pinpoint where the heat is being generated from if you think its something other than brakes.
Had a similar issue with my 82 C70 last week. I use my truck to run firewood. Was making a 150ish mile run, stopped to take a break and do an inspection and the rear most driver's side was smoking. I had a heavy load on it (about 35,000lbs) and I'm thinking maybe I didn't get the wheel bearing on that wheel adjusted tight enough. With it a little loose it was dragging on the brakes a little bit. We stopped and poured water on it every 25-30 miles, it was getting hot enough to steam, but wasn't a fire danger. I ended up rebuilding the lower wheel cylinder since it looked rough (had a bunch of junk in it... even though I've throughly flushed the brakes) and readjusted the wheel bearings. I haven't run another load yet (just getting done with it today), but I'm hoping that was the fix. Little bit different than your being a tandem rear axle and drum brakes, but maybe some ideas? Sounds like your is only single rear? For some reason I thought all C70s where tandem.
When you did the brakes, did you clean caliper mounts with a course lock pad. These are bad about not returning to center. Is the inside brake pad wearing more than the outside pad?