Marko, Here is my $.02 worth. I have the Reese Revolution and love it. I'm not sure what you are referring to when you say ""moving to the side when hitching up." If you are indicating that when the hitch engages the king pin the moves sideways instead of sliding into the hitch "V", I would say you are not lining up with the revolution. One has to be lined up with it in order for the "wedge" to slide into the "V" of the hitch.
On my RV, I could not move the pen box by hand from side to side until this past spring when I removed it to do the annual service on the bearing. The bearing was DRY. After packing the bearing and re-assembling, torqueing the castle nut to the 80 LBs, I can now with moderate effort move it from side to side by hand. If you can move yours
with little effort, I would suggest checking the torque on the castle nut.
Also, check the wedge. It has to be adjusted to the hitch.
Hope this helped