I've looked at those pictures a couple times, and frankly I can't see any "slight separation" between the fretboard (not neck -- dulcimers do not have a neck) and the top.  I don't see any "area that has dropped", either. 

If you must do something other than just play it, as Irene suggests, I would tilt the instrument 45 degrees on its side, and using either a fine squeeze tip from  superglue bottle, or a toothpick, put a couple drops of 'slow set' superglue right in the joint.

I say that this "slight separation" isn't enough to worry about.  Just enjoy playing her.   aloha, irene

Greetings! Some time back I picked up an Edd Presnell 6 string,the wood was in good shape but there were separation issues on the tail, crack on the back, yada, yada ,yada, have it nicely fixed up. I still have not tried to fix the seam separation right where the neck meets the top surface. There is also a slight indentation right where it is separated.

 2 questions,

 Should I try "lifting" the area that has dropped? And if so how do I do it safely.

 second, What would you suggest for gluing the seam without a big mess? I have considered an exact o knife or scalpel blade to clean up the separated area, but as to applying glue, what is the best, cleanest way to do that.

 Attached pictures, let me know what you think. Thanks everyone.