I've been a cabinet maker for 27 years. The best way to handle this is not by using Elmer's glue. Buy a better product such as Franklin Glue. Most likely at the age it is the glue was made from animal hide and if you can remove some it may help. Don't worry too much about it. If it took any of the wood from the sub-straight with it when it came loose, do not remove that portion.

Apply a bead of glue to both services and spread evenly. If there is any way you can borrow a three clamps you would have it made. You would then place the trim exactly where it came off and apply a little pressure with the clamps until it dried. If any glue squeezes out you can remove it with a damp cloth and it will cause no harm. Do not allow it to sit to long before doing that.

I wish I could see the cabinet so I could give you better advice. If you have any more questions, just post them and I'll watch for them.

You can do this with no problem. If you do use clamps, place a soft wood such as pine between the clamp and the piece you are attaching.

I sure hope this helps. If you have a concern, do not continue. That is a valuable item you have.