I would weigh the truck, to see the real weight is with the options that your particular has. Weigh each axle to see how the weight is distributed.
I regularly see Tundra's towing 27-29 ft TT's with one slide . I think you need to do as someone suggested already, rent a TT and see if you like it . See how it feels with your Tundra and go from there. See how crosswinds feel using your truck with a TT . You might hate it ! Then you can visit the rv.net motorhome section .:B
Re trailers dry weight....i would forget that number ! They post them to attract unsuspecting buyers to make them think they can tow a 35 ft TT with a Kia.
I have never towed or hauled any trailer or truck camper that was close to a dry weight and that was from the dealers lot, empty. The RV's listed options usually bumped up dry weight numbers by 500 lbs.
My dealer used portable scales to get real weights of his units on his lot.
***many potential buyers felt they had to downsize the rv or upgrade the truck and many others went ahead with the purchase and towed for years w/o issues. Im not saying its a good idea to do that, but people do it and i see those people all the time when they return to the shop for service. Still towing, still smiling and never had an issue regarding weight.
What i see some 1500 owners do is upgrade the tires . If not for weight reasons, they simply feel better while towing.
"My truck feels kind of squishy towing this thing". Owners have said, the stiffer sidewall of an LT tire, just feels better. I guess it becomes more of a factor as you get closer to the max ratings of things.
Your trucks tires also have a weight rating stamped on the sidewall. "P" rated tires might be your limiting factor and might have to be upgraded to an LT tire. If that's the case, make sure the rims (wheels) will match any tire upgrade.
http://towingplanner.com/Calculators/TowingPayloadEstimatehttp://www.rvuniversity.com/staticpages/index.php/Matching_Tow_Trailer_101