Summit,
Welcome!!!
Unfortunately as you have found out, dealers will tell you your tv (truck) can tow it, when in reality that may not be the case. It is very good you have found this sight before signing any papers!!!
If your plan is to have a tt delivered to a site for this season, then upgrade the truck before starting the road adventures next year, then the model that fits the family the best, and is the best value would be my suggestion to go with....
Regarding your current truck, understand that all passengers, cargo (coolers, firewood, bikes, etc), any accessories added to the truck (truck topper/tonneau cover, step bars, bed liner, etc), the wdh, and the tt tw (travel trailer tongue weight) all count toward the tv payload...... Just how much does the family weigh, with the dogs? Know that kids will only continue to grow, which means heavier passenger weight, resulting in less payload left for the tt tw.
As mentioned, do not go by the "brochure" dry weights!!! The dry weights do not include options, even those options that are "mandatory options". The tw alone is deceiving in that it does not include the weight of a battery or the propane tanks. Those alone will add ~120lbs for a battery and 2-20lb tanks or ~160lbs for a battery and 2-30lb tanks. And you haven't even loaded anything in the tt yet. Because of this a lot of members suggest using the gvwr of the tt and estimating the tw to be 12-13% of the gvwr. This will give you a rough estimate of the loaded tw. Understand you will want 12-15% tw for a safe, stable tow. To little tw can result in sway, which is not a good situation. Using the 12% or 13% of the gvwr, then adding in the family weight, and 100lbs for the wdh (some are lighter, but a round estimated amount), you can compare this total to the yellow payload sticker amount in the drivers door jamb. Remember, you haven't added the weight of any coolers, firewood, or bike that may rise in the truck bed.
Don't get scared or overwhelmed by this info we have given. Anything is possible, being informed will put you in the drivers seat when shopping.
Regarding the dealers you listed, I wouldn't go near Gillettes if they were the last dealer in Michigan!!! I personally know of two families (one is a very good friend) who have been screwed over by this dealer. I know the salesmans name for one incident, and for some reason I think it was the same one for my buddy, but that was over a year and can't remember for sure.... It's to bad to hear how you were treated at Haylett as I have heard great things about them.
Good luck, and ask if you have more questions!!!