After looking around in the attic, I noticed that the ductwork was leaking at nearly every joint. I have not done this before, nor did I stay at a Holiday Inn Express last night.

I HAVE watched a crew replace the duckwork in a rental of mine though. I watched very closely thank goodness.
This is how most joints looked before I started.
I cut off the tie wrap holding the flex duct.
Oddly, on this side of the house, each duct was sealed with duct tape. Even after 20 plus years, the duct tape was holding strong.
After removing the tape and straightening the tabs some, I removed the steel adaptor from the duct board.
At least at the beginning the joint probably did not leak as much. Since the tiny bead of silicone was coming off though it leaked now.
