I just finished a long search for a vehicle. Used cars.com and autotrader.com to help with pricing. For your price range ($10,000 if I read it right) you are going for a high mileage, older vehicle. Unless the 10,000 is just a down payment. If the vehicle has been taken care of high mileage should not be a problem. 95,000 miles on my (new to me) 1500 did not scare me. Check with dealers and test drive.
Ford F250s appear to be less expensive than Chevy 2500s. With Chevy a 2002 to 2008 with an 8.1L will pull 11,000# - 12,000# TT with a weight distributing hitch. 2009 to 2013 needs the 6.0L with 4.10 axle ratio. 2014 and newer had the 4.10 almost standard. With the Ford F250 you need a 6.2L and it appeared that most was able to tow what I needed. Towing capacity was checked with towing guides at trailerlife.com.
The trailer should have a sticker on the side or maybe in a cabinet that will give you the GVWR. You listed the dry weight. Just don't add more contents to overload the GVWR.
Have fun and don't rush it. Just my two cents worth.
