I have no problem towing my 21' trailer. It's 3800 lbs ready for a weeks camping(actually weighed on a scale). See the wintery pic of it in my library.
You will need to have a Class 3/4 frame hitch installed. You will also need a weight distributing hitch setup along with an antisway setup which you said you do have.
The biggest thing you will need to do is have an additional tranny cooler installed in series with the stock one. The trannys are a weak point when towing, and I have needed the extra cooler when in the mountains in hot weather. The one I have does not have the electric fan... it's just an additional cooler mounted up front. OD is best left off, unless you are in really flat country. The tranny builds heat really fast every time it shifts, and that includes going in and out of OD. I have a Scanguage II to monitor the tranny temps.
A brake controller is a must for safe operation too. I brought my rig home about 4 miles without one, and that was enough for me. Kind of scary on stops without it, and your 2001 has rear drums (my 2003 has rear discs). I have the Prodigy, and am happy with it. There is no wiring pigtail for brakes on our STs, so you will have to install it, or have it installed.
As was pointed out above, make sure all the required maintenance items are done. A tranny flush would be a real good move.
Gas mileage will be in the 10-12 mpg range... :sad:
You might need bigger mirrors too. I have found the McKesh mirrors about the best on the market, but they are not cheap. Google them, as they are only sold on line at one place.
You will know the camper is back there, but the ST handles it well, given the size of both rigs. I typically run about 60-65 with my rig. Remember, most trailer tires are only rated for 55 mph.
...and... :welcome: to the site!