I believe all of the half ton truck makes offer a truck that can tow that under $37k. It may not be a 4wd with all the bells and whistles though.
OP, look at the window sticker or "monroney" label as they call it. Look for things like tow package or trailer tow package and make sure it has the proper axle rating, wiring, and receiver hitch for towing. If it has a N/A(naturally aspirated) V8 or a turbocharged 6 cylinder, then it will be able to tow what you have. I would not recommend a N/A V6 or turbocharged 4 cylinder for your trailer.
The more gearing the transmissions has, the less relevant the rear axle ratio. Turbocharged engines can also get away with taller gear ratios(lower number) than N/A. For example, the 3.55 on my old turbocharged 3.5L Ecoboost work truck pulled the same loads better and had a higher tow rating than the N/A 5.0L V8 with a 3.73 I had before. In my opinion, I would not go lower than a 3.55 on a turbocharged gaser and no lower than a 3.73 on a V8 within that price range.