The previous poster gave you good advice. The entire dash has to be REMOVED in order to get to the heater core, and the freon has to be evacuated from the air conditioning.
I did this job several years ago on a Mustang (mechanically the same car).
I sold the car a month later, because I never wanted to do it again.
How do I change the heater core in 89 thunderbird?
Get a Chilton or Hines shop manual. They have details on how.
How do I change the heater core in 89 thunderbird?
Not a job for the faint of heart. A T-bird with AC the job is 7 1/2 hours by an experienced tech. Without AC a little over 5 hrs. A service manual will be a necessity. Changed a few of these. Glad there are not too many around anymore. Have fun with it.