Not a truck camper type myself, but when you said....
The dealer said it would fit onto my Dodge Ram 1500 Quad Cab just fine, and it should be able to carry it well. He suggested and installed air bags to beef up my suspension.
....the red flags went up. How do you tell when a dealer is lying? His lips move. And then you said....
However, when my husband drove it home, he said it made the truck sway quite a bit, and felt very top heavy. I'm very new to this (this is my first RV), and don't know if this is normal.
... and the sirens went off.
First thing is get the the gross vehicle weight rating of the truck. Check on the plate on the drivers side door. It should give it and both axle maximums. Now you need the weight of that camper. It may be somewere on the rig itself on a plate but it is better to actually weigh the truck and camper as loaded to travel.
Go to a public truck scale.* Tell the weighmaster that you want two weights and that you do not need certified weights. Pull the truck fully on to the scale and get that weight. Then pull the truck's front wheels off the scale, leaving the rear wheels on the scale and get that weight.
If the first weight, the total weight, is greater than the gross vehicle weight rating of the truck, there is nothing you can do. You have a mismatched camper and truck. It will be unsafe to operate.
If the second weight, the rear axle weight, is greater than the rear axle gross axle weight rating, but not by much, beefing up the springing and shocks may help. I would consult a shop that specializes in suspensions and alignments or a Dodge dealer that sells a lot of trucks. If the weight discrepancy is too high this condition may not be curable either.
If all the weights are in specs, then I would recommend looking into heavier duty shocks and maybe airbags. The 1500, like the GMC 1500s or the Ford F-150 are light duty trucks, with a compromise between load carrying and passenger car ride. A 2500/250 class of truck will be a lot happier with any camper bigger than a camper shell.
Hopefully actual truck campers will butt in here and give some advice. In any case you are right to be anxious.
[*Public scales are found at truck stops, terminals, contractors rental yards, and such. Look in your yellow pages, or, better yet, your local business yellow pages.]