I would say generally don't overstay. Some people say there's no problem but if you for whatever reason getting to trouble if the police you get arrested and deported....
You also get a lost stamp in your passport if you leave and the finest 500 baht a day. Oh that stamp doesn't look good for any other visa and any other country you may apply.... Depending on the country.
Why don't you take a week and go visit Cambodia? Then when you go back you get another sixty days
The biggest issue will be opening a bank account locally and transferring money as well as getting a rental contract for an apartment. Just finding a bank is going to be quite time consuming. Bangkok Bank is usually the best bet. Many other banks have restrictions.
As others said your best bet is to talk to the local immigration department about what you need to do. Asap.
I'd like to verify that at this might come up later that you can renew your passport abroad and still enter using the stamps in the old passport?
Is it really in the requirement to move over the Visa and extension stamps? I know I had to do it before my extension in country they just gave me the forms and did it at the same time.