The truck will pull 6000 lbs easily that isn't what you need to worry about. The thing that limits towing capacity is payload. There should be a sticker on the drivers door or door jamb with 2 numbers on it. One will be payload, the other is GVWR. The GVWR is the total weight of the truck and contents including trailer tongue weight. passengers and cargo. Payload is GVWR minus the weight of the truck when it left the factory.
My truck is rated to tow 7700 lbs. My GVWR is 7350 and payload is rated at 1621. That means my truck weighed 5729 lbs when it left the factory. The combined tongue weight, and weight of passengers and cargo shouldn't go over 1621. BUT.... My truck actually weighs 6000 lbs with the gear i keep in it, my fiberglass cap and myself sitting in the drivers seat. Now my payload is only 1271. A 7700 lb trailer will have about 1000 lbs tongue weight plus about 100 lbs for a weight distribution hitch and only leaves me 170 lbs for my wife and cargo in the truck..
For my truck about 5000 lbs is as much as I'd feel comfortable pulling. I don't need a WDH and with about 650 lbs tongue weight I'm left with over 600 lbs for other passengers and cargo in the truck.
You need to figure out how much weight your suspension is rated for 1st. You are probably right at the limit with a 6000 lb trailer, but could be over. You might have room to spare. I'd strongly advise taking the truck to some scales and find out exactly how much it does weigh. Any accessories you've added such as bigger tires, bumpers, winches, tool boxes etc will cut into available payload. A 4X2 truck will weigh less, as will a regular or extended cab. My crew cab is heavier and that does reduce payload.
3/4 ton trucks can handle more weight because they have more payload, not necessarily because the drive train is more powerful.
