One of the best things to do first, is to load up the truck with everything you would want to load in it when taking a trip, and with a full gas tank, go weigh the truck at a cat scale with your wife in the passenger seat! Include coolers (packed ready to go if possible), firewood (if you will take that), etc. This will give you your loaded truck weight, before trailer tongue weight. Now subtract your loaded truck weight from the gvwr (7000lbs maybe???) of the truck, to find available payload left.
A slightly easier, but not an exact way is to look at the driver door jamb for the yellow payload sticker. It will state "All passengers and cargo not to exceed "????"lbs". Subtract you and your wifes weight to start, then maybe estimate what you would load in the bed.
Another question- do you have anything additional added to the truck? Like a truck topper, spray in bed liner, step bars, etc...? All these items subtract from the amount listed on the payload sticker.
Your truck has the 6spd tranny, correct? Then the "5.3/3.42 say fuel economy" is not true! With the 6spd tranny, it is actually like a 4spd tranny with 4.56 gears!!!! So power wise, you are fine! I know this pulling a ~ 7500lb trailer with that exact combo in a '10 Silverado 1500 ccsb 4x4 5.3/6spd/3.42, though that has been here in MI, so not mountain towing!!!
The ONLY reason we now have a 2500HD is due to payload, which will be the limiting factor with your truck. We carried the coolers and firewood (when we take it) in the trailer due to actually being just over gvwr by about 200lbs. We were under the axle ratings though! We did have a cab high topper, spray in bed liner, step bars, which all "ate up" some of the available payload! So now the coolers/ firewood ride in the truck bed without any issues!!! Plus we are a family of four (lightweight for now as we have 10yro twins who are growing!)
One other thing to keep an eye on is the factory round tube hitch! On our '10, it would actually twist a little when hooking up the wd (weight distribution) hitch- hooking up the bars. That was with a little higher tongue weight (tw), almost 900lbs. Hitch was rated for 1100lbs I think???
Our old trailer was about 25', new trailer is about 30'. Didn't notice much difference between the two towing with the 1500.
Depending on the trailer layout you are looking at, I would estimate you should be ok with a trailer that has a "dry tongue weight" of maybe up to 600lbs. Loaded ready to camp this could easily be 900lbs or so. This is dependent on the accessories you have added to the truck, and what you may load in to the truck bed. If no accessories, then maybe up to 800lbs dry tw depending on what you will load in the truck bed. These amount are just guesstimates not know what you and your wife weigh, or what you will load in the truck bed!
Good luck!
Chris