You didn't specify the year of the truck. I will asuume it is the 88-98 body style because this is what I have and I know some numbers.
There are only two numbers that mean a hill of beans. The GVWR, GAWR, and the GCWR. The tow rating is a bogus marketing number but if you have a 350 and 3.73 gears, yours is 6500 lbs. The GVWR is the weight on your 2 axles, should be 6200 lbs. The GAWR is the weight allowed on each axle, don't exceed this one, the tire weight ratings as delivered from the factory will not limit you. The GVWR and GAWR are available on your door jamb sticker. The GCWR is the total combined weight rating. The GCWR is not easy to find, it isn't in the owner's manual or on your door jamb sticker. For a half ton with 3.73 gears and the 350 it is 12,000 lbs your truck dealer can verify this, it is recorded with your VIN at GM.
So your truck should weigh 5000 lbs empty but with you and fuel, my K1500 weighs 5800. Looks like you really have 7000lbs left in GCWR and 1200 lbs of cargo capacity. These weights require that you know the actual weight of the truck. Go to a scale somewhere. 7000 lbs trailer/1200 lbs cargo is pretty good.
A trailer needs 10-15% tongue weight to avoid swaying. 10% of 7000 lbs is 700. Life is good, you are within the cargo capacity even with the tongue weight.
Your class 3 receiver hitch has a max tongue weight of 500 lbs unless you use a weight distributing hitch system then you can go to 1000, see the sticker on the hitch to verify. You need a WD system with the spring bars.
The 7000 lb trailer is a typical car hauler and will weigh a good 1000-1500 all by itself. This leaves you with an allowable tractor weight of 5500-6000 lbs. This weight is safe and sane, legal, etc. Drive across the country if you wish.
Now you may choose to exceed your ratings. This is a personal choice. I exceed my GVWR but not the GAWR because I refuse be limited to 400 lbs of cargo capacity in a full sized truck. The more you exceed your ratings the slower you should go. I don't care if you drive ten miles from home with an overloaded setup, you are just as likely to have a problem in those 10 miles as you are in 1000.