As long as you have a valid state driver's license from the U.S. you will not be required to take a road test. You do not need an International DL, and there is no form to fill in.
I just renewed both my auto and motorcycle licenses last week at the Department of Land Transportation in Bang Chak, near the Bag Chank BTS station, (even side of Sukhunvit, set ~ 1 Km back, approx. opposite Soi 99).
Go to the nearest DLT office to your affidavit residence.
I used this local U.S. Embassy web-page to identify the necessary documents.
http://bangkok.usembassy.gov/driving_in_thailand.html
1.) The Affidavit of Residency, the is $50 or 1,600 THB. I made an appointment at 10:00 AM arrived at 9:00 AM and was out the door by 9:15 AM. (If you are otherwise listed on a house record (Tambien Baan) then you may not need the Affifavit.)
2.) Medical Certificate, 100 THB, available from any clinic.
3.) Original, and two (2) copies of your current driver's license.
4.) Original passport, and two (2) copies each of the picture page, your current visa, Departure Card, Arrival stamp.
( I brought photos but did not need them )
The process is:
1.) Check in, usually at a desk near the entrance, or straight ahead where they will give a queue number, the first of several, check your docs, have to sign any copies.
2.) Go to another desk where they do some data entry (I think).
2.) Response test, eye exam (depth of field, peripheral vision).
3.) Queue to get picture/license, pay fee.
If your U.S. license is for multiple vehicle types then you should get both licenses here, make extra copies of everything.
For my first licenses I was in and out in 45 minutes. Last week the DLT at Bang Chak was packed, it took ~ 4 hours most of which was the final wait for the picture/license. – I got number 256 while they were serving number 112! I left for two hours and returned.
I think I paid 505 baht for my auto license and 255 baht for my motorcycle license but these are five (5) year licenses, plus 100 baht each for the picture/lamination. I think the initial one year licenses are less expensive.