That'll teach me to write from memory. I actually found the original post:
Adobe - Adobe AIR for Linux
You have to download the AIR SDK, not the binary file to do it the way I described. I originally tried the "official" option of installing the binary and then using it to launch AIR files directly but had all kinds of problems, not the least of which was updating.
I posted this mostly for the sake of others in trouble who end up here looking for answers: I'm glad to hear it's working for you, Mark!