TOW VEHICLE: GVWR 7200 GCVWR 14000 MANUFACTURER TOW CAP 8300
FULLY LOADED TOW VEHICLE(people, cargo, gas, etc.) 6500
TRAILER: GVWR 7500 DRY WEIGHT 5800 FULLY LOADED (cargo, propane, etc.) 6500 LOADED TOUNGE WEIGHT 645
From the figures you're giving above, you only have 700 lbs of payload left (7200 - 6500). Typically, a 7500 lb trailer "could" have up to 1125 lbs of tongue weight (7500 x .15). Even using your "estimate" of 6500 lbs, you "could" have 975 lbs of tongue weight.
Realistically speaking, 13% would probably be closer to the actual percentage and would therefore be 6500 x .13 = 845 lbs.
However, you are showing less than 10%....and that ain't good. Light tongue weights will almost always result in trailer sway. If that is the mfg'ers listed "dry" tongue weight, you most certainly will have a lot more than that once loaded.
So, you are over your truck's payload capacity using any of the above percentages. As previously mentioned in an above post, forget the towing capacity and dry weights...for the most part, they are useless figures.
Speaking for myself, I would look for a much smaller (lighter) trailer.
Ron