I had this same problem and fixed it. My source of the rocking movement was the splined shaft with the plastic gear, not the worm drive. It was moving front and back inside it's white plastic enclosure. That shaft is contained from fore/aft movement by a rubber washer. That rubber washer degrades and allows the shaft to move front and back. No amount of tightening on the 12mm nut will fix this. You need to replace the washer. I used a garden hose washer that I trimmed down to fit inside the plastic housing.
Thank you I never took my seat apart because I have all of my white caps but I will look into the rubber washer. It's weird that it only slides forward and never backwards. I guess it will make sense when I finally pull the seat and flip it over.