I looked at the on link given that has the 2008 towing guide for your vehicle.
While it states 5,000 lbs, it also states MAX OF 500 lbs hitch weight which gives me pause. That 500 lbs hitch weight can easily be reached at 3,400 lbs of trailer weight at 15% hitch weight without WD..
Additionally in extremely small print it does state that the 5,000 lbs is for a BASE MODEL of your vehicle.
What that means typically is 2WD, no options (other than the towing option) and absolute bottom, lowest trim package.
Adding 4WD, higher trim levels and any other options WILL reduce the max towing capacity.
So, depending on the exact configuration of your vehicle you might find yourself considerably lower towing capacity.
Something else which is often overlooked is trailer manufacturers tend to rate the dry weight A LOT LOWER than they actually weigh.
Often weights of battery, propane tanks (and propane) along with other optional items in the trailer ARE NOT INCLUDED in the published dry weights.. Battery and propane can add an extra 150 lbs easily to the hitch dry weight..
I will really suggest you consider a smaller than 22 ft trailer and you WILL need to be looking at the LITEWEIGHT MODELS.