No, nothing special about SHA+ hotels other than the staff is trained to intake "sandbox" guests. Remember, SHA (Safety and Health Administration) is just a certification that says a hotel is cleaning correctly. The + means more than 70% of the staff are vaccinated.
You said "I've never seen a PCR test required for domestic travel." I'm not trying to be a jerk (though I know I'm probably coming off like one) just that the PCR test was a big deal for people trying to get into Phuket, since it cost more than the flight into Phuket.
As of now, I don't know. I think most places accept AKG tests. My buddy flys back and fourth between bkk and kk, no tests required for him, but he's vaccinated.