Before doing anything else like replacing tires I would look at your weight ditribution hitch first as all you mention is stabilizer bars which are not necessarily the same thing and with a properly setup hitch you should not be sagging on the rear mcuh more than 1/2". If what you are refering to as stabilizer bars are weight distibution bars then you need to find out what rating they have as there are numerous sizes.
Have a look at the label on your trailer to see what the GVW of the trailer is and that will help determine the size bars you need as the tongue weight should be in the 10-15% range. The best way to confirm the tongue weight is to actually weight the tougue with the trailer loaded and also weight the rear axle of the vehicle loaded and unloaded. Then the weight of the tongue + the additional weight added to the vehicle will give you the exact range the bars should be rated at.
For myself, I have a GVW on the trailer of 10500# and I run 1200# bars on my hitch on an F150 with Max Tow and I have no issues at all. My weight as calculated above is 1050-1100#.
It is well worth the time to get the hitch setup properly as that will solve alot of teh problem along with a good sway control system like Reeses Dual Cam.