Good Morning, The Koni shocks ( 8805-1018's and 1019's ) will help the ride quality and help reduce the swaying, however they will not solve the entire handling concerns . I would take a look at our "Stage 3" suspension "kit" , I would recommend at least a rear Auxillary swaybar and a rear trac-bar to help reduce the pushing and swaying. The Safe-t-plus steering control is also good to help with wander and rut tracking, as well as providing blow-out protection.
https://www.ultrarvproducts.com/Stage-3-2006-Ford-F53-Class-A-20-22K-GVWR-Handling-Kit
You can pick and choose the products from that list , you can also give us a call at 800-417-4559 and ask for Ken to go over the handling and solutions.