@Jamie **************
, You need to go to the local immigration office once you arrive in Thailand to apply for an extension based on tourism purpose. Detailed information about tourism extension has been provided by James above with that sheet. You can forget about the TM6 arrival/departure card you wouldn't get one unless the government issues a new announcement.
But, IMHO, the best and cheapest way to extend your stay is a Single Entry Tourist Visa, not an extension. SETV fee is cheaper than an extension inside Thailand, and you don't have to spend half to one day to visit the immigration office, because you will be stamped for 60 days when you enter Thailand. You have to apply now where you are, though. If you are still in the US, you can apply online.