You should not have any issues towing that trailer. you are well under its rating. 3700lbs dry should weigh in around 5k lbs ready to travel. have you weighed everything to make sure you have plenty of tongue weight? Are you using a weight distributing hitch? Its only 22 feet so length isn't an issue.
I`m thinking you have some tweeking to do with your setup, as you would with any new setup.
And yes the engine will rev with a load while going up hills, that's where it makes its power, nothing wrong with that as that's where it makes its power.