as a general rule yes you can exit/re-enter and get a new free entry stamp BUT if you're thinking of doing it by land use a service that takes you to the border and back because they have it set up so you can get out/back in without issue
You will not get an extension based on internship issued inside the country from an immigration office, not from a non-B or from a Non-ED <- which is what most interns have here.
There is no way to get a "short term" extension based on internship, not 30 days, not 90 days, nothing.. It's just not available inside the country
You either go get a new visa OR you exit/re-enter on a free stamp entry if you come from one of the 90+ countries and just intern on it until you finish your program
at least in Bangkok if you apply with the online 90 day report system the first time after you come in from abroad they reject you and this is in the body of the rejection email
totally okay to have a visa/extension/re-entry in an old passport. The passport itself is 'cancelled' when you get the new one but any existing visa/extension/re-entry is still valid
you just have to go to the immigration office where the extension was issued and get the stamps transferred. All the airport will do is stamp you IN on the new one and note the extension/re-entry was in the old one
just so you remember it's the VISA <- the PDF from the thai consulate's expiration date NOT the small entry stamp you got when you stamped in to thailand's expiration 😉
Many people mix those two up and end up losing their ability to get that "second year" entry stamp