Your 3773 pound maximum Pin weight sounds about right on your truck.
If your trailer has a 2900 pound dry kingpin weight, you could easily be pushing 4000 lbs. after your trailer's completely loaded, fueled up, the weight of the fifth wheel hitch and with passengers to add.
Unfortunately, you're max'd out on your single rear wheel truck, and your trailer is usually seen with dual rear wheel trucks as tow vehicles. Not much you can do about your truck now, other than put a set of air bags on the rear suspension.
If you're towing in flat terrain, you'll probably be okay. If you were towing mega miles that included mountain driving, trading for a dually would be warranted.