The truck can easily handle the amount of weight quoted, but realistically you will be looking at much higher weight, probably in the range of 3500+ pounds when ready for a trip. Good tires and brakes are key, and an exhaust brake is very helpful.
I hauled my Bigfoot 2500 slide in camper all over America, many tens of thousands of miles with no problems. My truck was a dually though and the training wheels do give more stability and payload, but I would not be afraid to haul it with a 3/4 ton, I would just drive slower and approach curves a bit more cautiously.
Useful tips for enjoyable truck camping: travel with tanks mostly empty and fill up with water near destination - saves a lot of weight. Keep heavy stuff like drinks and canned goods as low as possible, in the basement if your camper has one. Putting heavy stuff in the cabinets raises center of gravity and a top-heavy rig will drive squirrelly. Put some of those heavy rubber horse trailer mats in the bed of your truck, it will help reduce the shifting around of the camper. Yes the camper can move around, particularly in strong crosswinds it can shift to the side and leave your tie downs loose. Good idea to check tie downs every time you stop.
On the camper, be very careful with the jacks. They are surprisingly stout in spite of the stick-thin appearance, when the camper is raised up for loading it can look alarmingly unstable but as long as you take your time and keep it level you will be fine. However the jack attachments are usually screwed into wood and I have seen truck campers with wood rot where the jack attachment screws were pulling out.
Really the most important thing is to drive a bit slower than normal. I found that my rig handled wonderfully at 65 on the interstate and at 55 on curvy mountain roads while getting very good mileage (often got 15+ mpg at those speeds). At higher speeds the handling becomes noticeably more "busy" and the driving is much less relaxed and enjoyable. 75 on the interstate is easily sustainable but 2nd Gen Dodge "steering wallow" will give you a heck of a workout. I learned very quickly that keeping the speed down translated into a MUCH more enjoyable ride. Took my rig up Pikes Peak a number of times, all over the Rockies and lots of passes and steep gravel roads, and the rig handled it better than any other RV combo I ever had.
Truck camping is awesome, have fun and be safe --- and watch out for low bridges and tree limbs :-laf