I read through the forum and cannot find my exact issue, so I apologize if this question has come up elsewear. I recently got a 17' TT Forest River 17bhse, towing with a 2015 f150 with a class 3 hit...
Your F150 should not be having these issues with a relatively small trailer. Your hitch setup is what I would focus on. 600lb tung weight should not get you into trouble either. I’d also look at your trailer tire wear and axel alignment. If its possible, have a friend tow the trailer in their truck, using your hitch as see what the result is.