Antique carved screen. It would be in a closet being protected if my couch was moving and hitting it. If it's worth enough to care about anyway. Things that have monetary value or mental value due to what it is and where it came from are things that are not allowed to be hurt. They get put up and cared for.
If you can't find a solution to the couch moving you need to put the screen away until you move or get a different couch, but any couch is going to slide on wood floors.
A couple of things.
Folding stuff allowed those folds to move. You only put one layer. That's why this option isn't working.
Put an area rug down in the center of your living room and put the couch front legs on the rug. That should anchor it some.
The suggestions below are good ones.