With pulling your +6K lbs trailer, turn the Tow/Haul mode on and leave it on. It will keep the engine in the torque band better reducing lugging. It will also help with engine braking keeping the torque converter locked up, using the cruise control helps with this also.
Don't freak out about the engine turning higher RPMs as there is only three options if you want lower rpms while towing: drive slower, buy a diesel, or just don't tow.
Turning the T/H mode off to save fuel might result a couple of tenths mpg but that will be quickly diminished when the engine pulls timing due to the engine lugging which lowers the power output. To compensate for this you'll step on the accelerator more so where is the fuel savings?