If it is a 5300 (is that what 5.3L means?) with a 3.73 axle ratio, then my owners manual says I can tow 7700 lbs max trailer weight with 13,000lbs GCWR. So is that not what that means? My family of 6 weighs in at less than 800lbs.
So Donn, what does very little towing capacity mean?
You need to be more concerned with the trucks payload rating than the tow rating. You'll exceed payload before coming anywhere near the tow rating.
Check the sticker on the drivers side door frame. It should give the payload rating or say something like " weight of passengers and cargo should not exceed..."
From that number you need to subtract the weight of your family and any stuff you would carry in the truck. What you have left is the max weight available for the trailer tongue weight and hitch. An common way to estimate the trailers tongue weight is to take approximately 12% of the trailers GVWR. Ignore factory or brochure DRY weights.
Example;
Payload = 1500lbs
subtract 800lbs for your family = 700lbs
Subtract the weight of the weight distributing hitch that you will need (~75lbs) = 625lbs left over
12% of a 5000lbs loaded weight trailer = 600lbs so you'd want a trailer that was no more than 5000lbs when loaded which would be a trailer with a dry weight in the area of 3500-4000lbs.
Your tow rating is also reduced pound for pound by any weight added to the tow vehicle beyond 150lbs for the driver.