26' Camper Sway Issue

Also make sure the trailer is level to very slightly nose down when running down the road. Nose up will also cause sway. 
Even if the tires are fully inflated, some are softer side walled than other, this can cause issues. OR they may even be overloaded for capacity. Or heaven forbid, someone put P=Metric tires on it vs ST or LT rated tires. 
Since you said it was bought used......if the axels are V'd, that will cause sway as the axles with try to turn on you.
Look at the weights side to side. Too big of a difference in weight on one side will cause those tires to have a smaller loaded diam, also cause a trailer to want to turn similar to the v's axles. My states laws say no more than 500 lbs side to side difference when loaded or you are overweight, finable and they can make you move items from one side to the other to make it so there is less than 500 lbs per side difference.
Movement side to side IIRC is 12 or 18", you will get a red tag no more driving down the road until the sway part is stopped. 
I am of the opinion; ANY trailer should be able to be towed without sway bars at legal driving speeds. Which to me is 60-65 mph. Granted some states, not many allow up to 70+, most are 60 or below towing speed limited. Then put the antisway bars on for added protection. Don't do JUST the bars, Hensley hitch etc to stop the sway, your trailer still wants to sway, so solve the reason(s) that have been mentioned which is the root cause of the symptom, ie sway is the symptom! not the cause. 
My 02 for what it is worth.

Marty