Along with cork tape for the job you'll also want to buy some foam tape to replace your now non-existent old foam on/in the heat/ac box.
I have done this job twice now - if there is any reasonable way, do try to save the existing evaporator; you can flush it out etc. Once you're removing the dash you'll wish you had salvaged it.
The first time I did this, I removed the entire dash and entire box. The latest time, I removed the dash and only half the box, leaving the heater core in place and connected to the cooling system saving a ton of time that would otherwise be spent draining, refilling, bleeding, etc. Be careful removing your dash & bannicle, if it is old and brittle you may crack things, you can also tear up your A-post trim removing the dash if you're not careful. If the dash is old and sunburned, now is the perfect time to fill cracks & spray them down with vinyl paint and make them look new.
Cork tape on amazon: https://www.amazon.com/gp/product/B0049MM5BM
Foam tape on amazon: https://www.amazon.com/gp/product/B00SYQ3VNA
There is more than enough of each tape to complete the job.