This video makes me think you need new plastic clips (or the door is broken). Clips on the freezer have the same part number.
I'm not sure if those heavy, decorative wood panels are Dometic or Forest River -- I suspect FR as I can't easily find them on Dometic parts lists. My refrigerator does not have them and the stainless front panel has never moved -- guessing 'cuz it weighs very little.
I'd think industrial contact cement would keep the wood panel in place. Nothing behind it so no reason to remove it.