All of the above (check tires and loading), but, I am a firm believer in sway control devices. You are towing with a light vehicle and an even lighter big box attached to the back of it. While your Trailblazer can handle a popup or utility trailer just fine with nothing added for sway control, think of the "sail" that you just added to the back of your tow vehicle. The air buffeting off the front of your Trailblazer, colliding with a 7' tall, 7 1/2' wide box is enough to start instability. Add any kind of cross wind and you'll surely have the tail wagging the dog.
If you don't want to spend or need the utility of a weight distribution hitch (most all of them have "built in" sway control), then, at the very least, invest in some add-on anti-sway bars. These can be relatively cheaply bought, and while the won't do anything to transfer weight off the rear end of your Trailblazer and transfer it to the front, they will reduce the likelihood of wind induced sway on your setup.
IMHO, anyone towing a full sized travel trailer, even the short ones, should invest in a good weight distribution hitch. You can tow without one, but, it will save wear and tear on your vehicle by balancing the load front and rear, it will aid in stability by putting some of the weight back on the steering gear, transfer some of the weight off the hitch and back on the trailer's tires lightening the total perceived loading on your vehicle and, equally important, most of the better ones (Equal-i-zer, Reese WDH, Blue Ox, etc.) have built anti-sway bars. Most folks towing small travel trailers are also towing with small vehicles. The balancing and over-all weight distribution are equally important when everything is scaled down. I need far less "sway control and weight distribution" when I'm towing with my Ram 2500 HD CTD than I do when towing with my smaller and relatively lighter (when compared to my pickup truck) Jeep Commander and the Jeep Commander with Hemi and 3 tons of weight is a superb tow vehicle. The Jeep and 210WBS are a good "mate"...size and weight. The truck is actually bigger and far heavier than the 210WBS and its sheer mass tends to keep the trailer "in line", but, I still use the weight distribution hitch and built in sway control (Equal-i-zer).
My 2 cents, your mileage may vary...
2 Cats; J-Lo and Ragamuffin :R
2014 Thor Tuscany 40RX DP
2011 Ram 2500 Longhorn CTD HO
2011 Keystone Cougar 318SAB (now gone)
2008 FunFinder X 210WBS (Sadly gone)