I just replaced the contactor and the AC is running perfect (and very cool, Yay). The coil on the old contactor is burnt and almost melted. I believe now I know what happened from the beginning that caused all the problems.
When the unit stopped working last Sept, the control board must have had a short that allowed line voltage to get to the 24v circuit and burnt the transformer and also fried the coil on the contactor. Once I replaced the control board and transformer there was no need for AC at that time and when heat was called for, the furnace worked perfectly. Once the season changed and I had to use the AC, the system would not work. Since I replaced the control board with a new model that is fused on the 24v circuit, every call for cooling would send the voltage to the fried coil and it blew the fuse due to high amps. So, replacing the contactor eliminated the failed coil and now the 24v circuit is working properly.
My thanks to all who have contributed to identifying and fixing these problems and helping me understand the chain of events. My total cost of repairs since Sept is a bit under $200, including the new control unit ($100), 2 transformers ($17 ea), 50 ft of T-stat wire ($9) and a new contactor ($38).
Have a wonderful (and very cool) summer.