Fred, have you done the test(park on a hill...)? You may want to try changing the working fluid, most likely 20W50 motor oil. The 'VI' used in the oil can break down, and the viscosity at temperature may be too low for effective use. Some units are equipped with filters, so I would check if you have done, and perhaps consider replacing if equipped along with changing the fluid.
If worn out completely, the input pulley on top generally has a fan built in. If the trans is covered with clippings or debris, it can hinder airflow used to keep the temperature within reason. The clippings etc should be removed on a regular basis if there is a buildup. It might pay to check the linkage for wear and looseness as that will affect control, and getting the unit into actual 'neutral', where it is not dead-head pumping while sitting still.(causes heat)
When I had a hydro, I used the shift control lever to slow down on hills rather than the brake pedal. If you do use the brake pedal, you also are disconnecting the power input to the trans, and it is no longer pumping fluid... again, could cause overheat.
If you get the mount dimensions and the shift control linkage setup, you should be able to find a substitute trans that would be a mostly 'bolt-in' replacement. (if you cannot find an exact replacement) Many have the same mount dimensions and the same linkage, making swapping in another unit/brand not super difficult.
tom