According to the current Dodge website Tow capacity calculator, a 2008 1500 Quad cab,2WD automatic (5 speed) with the 5.7L Multi-Displacement engine & 3.92 gears is rated to tow 5,500 lbs. I could not find any MDS engines rated anywhere near what you stated for your 06, but the standard (non MDS) 5.7L can get up to 8900 with the 4.10 axle. TL does indeed show up to 9100 lbs for a 2006 5.7L, but it make no mention of MDS vs non-MDS engines. The Dodge site implies the MDS engines result in greatly reduced tow capacity.
So, what cab style (regular, Quad, Mega) and trim (ST/SLT/Laramie) and transmission 4 sp auto/5 sp auto/6 sp manual) do you have?
And none of this says anything about tongue weight - your trailer could easily overlood the 1500's payload or rear axle GAWR, so hitch it up and weigh it before going anywhere.
Dodge ought to be able to give you the GVWR, GAWR and front/rear GAWRs for your vehicle by looking up your VIN. Ask a Dodge deaelr to help you. That will give you a pretty fair spec on it, though travel trailer towing capacity is sometimes less than GCWR-GVWR. Basically, though, if you are more than a tad under the GCWR you should be OK.