1962 Ford Thunderbird Convertible - Roof Problems

nice car- getting the power top to work correctly on a 50 year old car is a zen exercise, indeed- if it has not been worked on before, it probably needs one new EVERYTHING- that is a classic, and for a wiring diagram and advice you should already be a member of a classic T- Bird forum- having said all that, it sounds like the motor is working and getting power- get your test light out and check the switch, and the relays-

It sounds like it is the limit switch. on the passenger side about a few inches from the screw down, there is a limit switch/ button if you get something to push the button down it should satisfy and go to the next relay.There is a lil metal tab or trigger that gets bent upward after time or getting bumped. They bend upward and it doesnt satisfy rhatt limit switch so it keeps running , not knowing the top is already down. push the button down and it should work , if not the switch needs ohmed out to test it and then the relay