Over 4000 pounds, 585 hitch weight.  I wouldn't go w/o sway control for long but one trip you may be ok.  You could always adjust two things if you do encounter sway - depending on which tank(s) are forward of the axles you could load them up to add more tongue weight, which will help with sway if you have it.  So if the fresh tank is forward, fill it up.  If the gray tank is forward, fill it up.  Towing speed is another factor, and if you encounter it you may have to resort to coming home via back roads to keep your speed below 50 mph.  Another factor is tire inflation, so keep the camper tires at their sidewall max rating and your truck tires close to their rating (I've found I get a better ride if I keep them several psi below their max).

As for weight distribution, you need to check what the maximum hitch weight is for your vehicle.  If it is less than 600 pounds you'll need to use a WDH.  Remember that 585 is probably the hitch weight before all the extras are added like the propane tanks, A/C, microwave and anything else that the manufacturer lists as "extras".