IMHO, I highly suggest you take your car to a good transmission shop, one with mechanics who are familiar with these old transmissions.
You may need what I call, 'a transmission tuneup'. Over time, bands wear and tolerances change. These are easily adjusted AFTER the filter is changed and new fluid is introduced.
Do you have a Shop Manual for this car? If not, buy one. Most all the vendors have them for sale and they can be found all over the internet. This book is the single most important guide you will need for your car.
You have come to the right place for a host of experienced mechanics who have a passion for Thunderbird and other Ford cars. I suggest you become a Squarebirds.org 'paid member' for more in-depth resources and other great benefits. (It's well worth $20.) - Dave