it shouldn't
You could remove the ABS fuse but the ABS dash light will remain on indicating the ABS system is not functioning.
Simply get it plugged into an ECU reader as it is probably a fault that the ECU is rembering, if not get a manual for your motor and look for an additional sensor.
add coolant
The #1 cause of cold air from the heater when the car is stopped, or the engine is idling, low engine coolant levels.
There is a CEL, Check Engine Light, on the 1989 Thunderbird.
Heater core (radiator) needs to be cleaned or changed
The brake light fuse on a 1987 Thunderbird is located on the back of the headlight switch. It is not in the fuse box.
Low coolant, stuck thermostat, blocked hoses.
It could be from many areas such as the water pump, the radiator, radiator hoses, heater hoses, heater core, freeze plugs, intake manifold gaskets, etc. Take a good flash light and try to locate the source of the coolant leak. If you have no luck that way take it to your local garage and they can pressure test the coolant system to force the leak.
not likely. you will more than likely have a coolant leak in the car that ends up with coolant on the passenger floor. pressure test the system and look a for leak. the only way the check engine light would come on is if it had a coolant pressure sensor. most cars don't, only sweet expensive rides
where is backup switch located on a 63 thunderbird
Check your radiator coolant level. On some cars, low coolant will let air bubbles into the heater core which can result in intermitant cold air bursts.