Make sure all your tires, truck and trailer, are aired to the max, indicated on the sidewall of the tire. It will help a little to keep them from walking.
To be honest though, you really need more truck. I pulled a 4500lb empty weight trailer with a half ton Silverado 5.3 with tow pkg and no matter what I did it still bucked up and down and swayed when in heavy traffic or any amount of wind. When a semi would pass, I would really struggle from the sway and pull. I found a deal on a 3/4 ton Silverado and all my problems were over. I see you have a nice truck, and may not be interested in anything else but you would be much safer and more relaxed with a heavier truck.
The old saying is, you can never have too much truck.