Yes your truck will handle close to 6000 lbs. We usually recommend staying under the max, especially with a fairly small displacement engine like the 5.3, but you've accounted for most of the extra loads that people typically forget about, so you are probably ok up to about 5700 lbs or so.
As for the trailer dry weight, don't get trapped by that. The trailer always ways more than that when you get it form the dealer. Full propane tanks is one extra and there will also be some water in the bottom of the tanks, even if it says empty, and 6-10 gallons in the water heater, water lines, etc.. Figure at least a 100-150 lbs over the dry weight and more if there are any factory or dealer installed options. You need to be looking at dry weights somewhat under 5000, e.g. the Freedom Express 24.