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.