Your T Bird HVAC operates with a dual heater control located at the bottom right hand corner of the radiator. The control is subject to failure on our vehicles due to the age. What you describe is the usual initial failure of the dual heat control valve.The heat cool functions work via control of the AC and heater combination. What you describe is a standard failure of the dual control valve, however, the AC should also be checked for full charge. If you check the temp of the heater lines coming from that valve when the condition you describe is occurring, one line will be hot and the other cold.
First just go out to front of car and open the hood.
My dual coolant control valve went bad and it was closed. I had no heat on either side. I tried banging on them but it didn't work. I finally had to replace itFirst just go out to front of car and open the hood.
Look down the left edge of fan shroud under the radiator hose to see the Coolant Control Valve (CCV).
Take a long screw driver to tap on top of each CCV relay and Viola!
Otherwise, you will have to replace the CCV. After the second time doing the replacement, you will see why just tapping the tops will be less trouble.
Keep us informed.
There is a notice at the top of the page for 2005 owners that goes to this ebay link for the best 2005 Service DVD- https://ebay.us/6vcMFR
Change the heater control valve for the same problem and it still has the problem. there air air discharge temp sensors for both right and left sides which feed the computer. They may be the cause but it looks like we have to rip the entire dashboard apart to get to anything in there. If you find the fi please POST for us all.