Let me give my qualifications here. My first career was as an ASE certified mechanic specializing in brakes, suspension, and air conditioning systems. Admittedly it has been a LONG time since I spun a wrench professionally, but the basics are all still the same.
Towing ratings are based on a LOT of factors, engine, transmission, differential, brakes, even frame. While payload capacity can be improved through helper springs, towing capacity is pretty much limited. Yes booster springs / air bags will help a little bit, but not by enough to really matter. If you need more than your 1/2 ton can handle, look for a 3/4 or 1 ton, or look to another MFG with a half ton that offers the tow rating you want. They are out there... I am not certain, but I think the upper limit for tow ratings on current half ton trucks is around 11K lbs (Ford F150), since I am not a Chevy guy I can't state what they run nowadays, and I think the Dodges still are in the range of what yours is... There are of course the imports, but hey, let's try to stay American shall we?
Now to the point of the load you are wanting to haul. Let's start off with, how much weight are you actually trying to haul? How heavy is the trailer? How heavy is the average log you are trying to haul? Do the math and see what your truck can pull safely without modding it. You might be surprised. I borrowed a friends 20ft dual axle landscape trailer (850 lbs dry), loaded it up front to back, side to side, 3' deep (its's as high as the sides went) with cut up pecan limbs and scraps. (cut off branches, leaves and the like)Total weight ~6500lbs including trailer (access to DOT scales varies by state...). You may be shocked to see you are under what you think you are, then again, you may be over as well!
You may want to check out the tables over HERE
to see what various wood species weighs at what moisture content...