Just carry both passports until the visa expires. A visa is still valid in an expired passport, but can't be transferred. Then when you get an extension of stay, you can get it inked into the new passport.
It is a 60 day tourist visa, not a 3 month tourist Visa. When the Consulate issues the visa, you must enter the country before the Visa expires, anytime within the 3 month period. You will be stamped in for your 60 day visa on arrival into Thailand. Once in the country, you can apply for a 30 day extension of stay (not a visa).
You should go well before your due date. At least a week or more, to be on the safe side. Leaving it until the last minute is asking for trouble, if there any issues with your documents.