Face it you're not the first foreigner to overstay in thailand.
Just go to the immigration office (don't worry about dressing up or acting all contrite) pay the overstay fine and apply for the 30 day extension.
As
@J*
, pointed out there's no need to wait until the last day because they ADD the extension on to your current stamp's expiration.
I bet you'll look at your admitted until stamp from now on though
Good Luck, report back and let us know how it went.