No problem to do the transfer at Suvarnabhumi Airport as a tourist on visa exempt.
You queue up as per normal, they'll send you to a special desk where you're stamped out in your new passport while some stamps from your old passport are transferred to your new passport for reference. They also link your passports in their system.
If you get your new passport before you extend your visa exempt entry at immigration, you need to do the transfer there and then, as they won't put an extension stamp in an invalidated passport.