Is there a vent of any kind on the rear end?
Yes there is. There's a plastic
VENT TUBE on top of the pumpkin on the passengers side of the car. Make sure it's there and not plugged or crimped.
HERE'S a shot showing where to look. (Note that the vent is missing in this picture and only the hole is shown.)
The axle will get hot after several miles of driving, so that's fairly normal. How hot is it getting? You should be able to touch it without getting burned, but it WILL be hot!
It also sounds like you need to replace your axle seals. After 40 some odd years, they go bad and leak. As a matter of fact, I just replaced mine a few weeks ago.
Ed