hi skibumdog. good to see a new member. Kirk does all his damping with his right hand. His use of heavy strings and a heavy
slide seem to cut down on the overtones. I bought a heavy slide and strings something like his and found that the overtones were no longer an issue unless you really want to play super clean tones. I just had a thought, maybe tuning in dropped D is less prone to overtones. Maybe the other players here who use other tunings could answer that question. Using a dropped D slide doesn't hurt either,fewer strings covered.
allthumbs