Sarah, that's another thought -- when you rent the trailer, make sure that the brake controller is properly adjusted. If you are not confident that the rental agency knows what they are doing, just Google "how to adjust trailer brake controller." It's not a hard job.
Some folks will say that you should be able to reach down and trigger the controller manually in case of a skid or a bad sway event. Frankly, I doubt that I would have the presence of mind to do that in an emergency, but apparently some folks can. If that ever happens, I will just have to trust that the controller will do its job while I focus on the steering wheel.
I should add that in 13 years of towing, and well over 100,000 miles, I have never had to make a panic stop. I try to anticipate bad situations and leave enough room that if another driver makes a mistake, I will still be ok. So far, so good . . . .
I should also add that although you will find towing stressful at first (judging by my own experience), it gets easier and easier. It will never be as easy as not towing. Towing requires an extra layer of vigilance.