Does anyone know if you are required to transfer to a new passport (new passport received while in Thailand) at the airport, or can it be done at a land border? Thanks...

Without knowing what it is you have in your passport, no-one can answer that question

Tke both old and new passports to your local I/O and ask them to make the transfer.  if your new passport was issued in Bangkok you might get a letter from issuing office asking IO to make the transfer

There is no visa to be transferred.  Just an admitted until stamp.

https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1156569/Information_Note_Visa_transfer_to_new_passport.pdf

You can leave by land without having the entry  stamp transferred in advance. However, at land crossings, there tends to be a designated official who handles the transfer. At a small crossing, it is not unusual for that official to be unavailable for hours, and you can be in for a long wait before being able to leave.