The ONLY way that could remotely make sense is if you're leaving by land, because you cannot change passports by land border. But if you're leaving by air there's literally no reason to do what you are talking about and it could only create problems for you.
Stamp into Thailand on your US passport. Stamp out of Thailand on your US passport. Stamp into the next country on your Russian passport.
it was designed to allow one extension per entry. But that was created by the MFA and immigration doesn't like it. So just leave and return bypasses that and gets you a fresh 180 days.
each immigration office will have different requirements. Generally your passport, TM30, passport photo and application form. Some offices charge and some don't.