Most 97s need a new ecu with the updated valve body. You will know the ecu is not correct with the replacement trans if there is a large delay into drive or reverse. Updated control module will correct some shift complaints, but not all. You might want to bring it back to the shop with the harsh downshift complaint along with hanging in gear issue.

Regarding the snow tire question, the computer does have adaptation and may take a while. If you unplug the battery, you really have to leave it for an extended amount of time to clear memory. There is no "version coding" in the trans ecu regarding to tire size. The only thing the ecu is coded for is based on your VIN number.Hopefully I helped a little and didn't confuse you more.