I had much the same problem with a couple of the wood strips on my '82 300D/4.3, and put them back on with contact cement. They've been back on for over 2 years now, so it must have worked ok. I don't remember what I used to clean up the pieces with, or if I even cleaned them. As I remember, there was some kind of slightly flexible material on the wood, and I just left it there and used a good coat of contact cement on both it and the surface it was going to stick to. As I said, there may be a better way, but my strips are holding well. One of them was the strip on the glove box, and the other was the one on the left side of the dash. Be sure to let the contact cement dry slightly before sticking the pieces back together, and be sure you get them right where you want them, as they won't move once they contact each other!
'82 300D/4.3L V6