The truck's axle and wheel rating is more than 8,900 lbs. so your are over its rated capacity with more than 5,800 lbs. in the bed. Payload is based on weakest link and usually it is the tires which are easily upgraded.
What is need to handle the weight of a camper varies by truck make, model, and year. The frame on a 2011 GM truck is substantially stiffer than on a 2010 GM truck. Same applies to the 2013 Ram 1-ton and 2014 3/4 ton trucks compared to earlier Ram trucks. As was already recommended, first try your truck as is and evaluate its handling.
With the extra weight the stock shocks are likely to be the first item that needs to be replaced. The Bilstein 5100 or Rancho 9000XL are commonly used.
Air bags are a mixed bag so to speak as they help with the load carrying but tend to hurt handling. Look for threads on how people have tuned their rigs to minimize the downsides of airbags.
Some trucks with lots of frame flex will benefit from a heavy duty rear anti-sway bar. Some benefit from adding one to the front. It makes no sense to generalize though as some trucks do not need them. My 2011 GM SRW truck handles a 4,000 lb. camper setup with the factory stock front suspension and only the addition of SuperSprings to the rear.
A DRW truck is going to be different and although the outer tires can provide more support for the camper the outer tires also cause more frame flexing than with the same load on a SRW truck.
