Technically it's not your responsibility to report where you're staying (the landlord/hotel owner is the one who will have to pay the fine for not reporting) but you won't be able to extend your visa without it, so it's in your best interest to make sure that the owner of wherever you're staying has reported it.

@Ann ***

I think so, but as long as you have one you should be fine. They've never asked me questions about it. The only time I had trouble was when I didn't have it because neither me now my husband (who's house I was staying at) knew about it 😂

I'm not sure about moving the stamp to your new passport, but as for entering the country again on another 30 day visa exempt, there shouldn't be any problem. Coming into Thailand by land on bisa exempt is limited to two times within the calendar year, but if you fly in you can do it up to six times (at least for my nationality).

If you want to get a 30 day extension, you need your TM 30, passport, one photo, and there are 3 forms you will have to fill out at immigration. You will need to submit copies of your passport and TM 30 as well.