The F-Superduty had 16" wheels and tires so that is not a tell tale of what you have. If it is a superduty, you will have a 2wd monobeam front end that uses leaf springs, 10 lug rims; hydroboost, rear disc brakes and a separate emergency brake attached to the rear of the transmission. If I am correct, 4:88 and 5:13's are the common gears for the rig and a Dana 80 is indeed used. The gross vehicle weight rating is in the neighborhood of 14,500lbs.
An F350 on the other hand will have a twin I beam front end that is coil sprung in the OP's 2wd configuration, 8 lug rims, vacuum assisted brakes, rear drums with the traditional E-brake set up, and most likely come with 4:10's. F 350's GVWR should be in the 10-12,500 range depending on how it was optioned.
The title should be a good indication of what the truck is, however, a better look at the hardware will be the true tell tale of what we are dealing with. You may have an F350 that had an F-Superduty rear axle swapped into it, giving you 8 lugs on the front and 10 lugs on the rear. Also, as Calvin has attested to, the Superdutys are a mean animal and can literally tear the sheetmetal apart so it is possible that an F350 cab has been transplanted on a 450 chassis.
With the weight that you are carrying, I would leave the 5:13 gears alone and hope it's a 450/Superduty, especially if you are toting a trailer behind a 15,000lb truck. My F350 grossed 15-16,000lbs towing a toyhauler to the rally and back, I wanted more than the 4:10's had to offer. I would go to 4:56's if I were towing on a regular basis. Also remember, the 5:13 gear set is allowing your transmission, engine and driveshafts to live an easier life. These engines were designed and tested to run on the governor, they aren't too keen to being lugged around.
Please dig into your rig a bit more to see what we are really dealing with. As for your gear set: mark your tire with a piece of chalk and do the same with the drive shaft. Get in your truck and go forward enough for the tire to go one complete rotation and have your helper count how many times the drive shaft goes around. A bit more than 4 times would be 4:10's, 4 and a half times would be 4:56's, a hair over 5 times will be 5:13's. Good luck and keep us posted.