Let me re-word this so that I can make it more clear.
US passport holder arriving into Bangkok later this month and staying in Bangkok for two full months in an Airbnb.
If I want to take a quick trip to Chiang Mai or Pattaya for a couple of days DURING that two months in Bangkok, and stay in a hotel in Chiang Mai or Pattaya, do I keep having to ensure a TM30 is filled out every time?
TLDR : Answer Summary
A US passport holder staying in Bangkok for two months doesn't need to worry about filing a TM30 unless they plan to visit immigration for something. The TM30 is the responsibility of the property owner where one is staying and is relevant only when interacting with immigration. However, it's important to note that the standard visa exemption is for 60 days, and overstaying could lead to penalties.