Dave, your 1960
did not come with an a/c originally, at least not from the factory. That point is important in that factory installed a/c units were riveted to the outer skin of the trailer, and would require a lot of effort to remove. Since yours didn't come that way, it is almost certain it has the standard RV install, which is clamped between inner and outer skins, making replacement a lot easier.
Inside you will find 3 or 4 bolts from the lower unit up through the upper unit, clamping the shell between the two halves. Remove the 120 volt wires from the lower unit, unplug the connectors from the lower unit to the upper unit, and unbolt the lower unit. It should fall onto the floor. The upper unit should pretty much lift off the roof, with a little encouragement in unsealing the unit fromt he roof (probably Vulkem or [shudder] silicone). The upper unit weighs about a hundred pounds, so exercise care lowering it to the ground, and lifting the new one. Clean the old sealant off the roof, apply fresh stuff, lower the new upper unit into place, and bolt up the lower unit. On the Carrier, the short bolts go in the front. Hook it to the 120 volt wiring, install the plastic cover on the bottom, and turn the unit on. If you did it right, you should be rewarded by cool air.