I remember something like this. I think it was Tam or Drifter. Not sure.
Anyway, remove the shift handle. There'* a small clip you have to pull out on the forward part of the shift handle. Use a small flathead screwdriver to remove it. Lift the handle off. Use a penetrating lubricant in between the shift 'shaft' and housing. While you're in there, lube the linkages and notches with a light oil. After the penetrating lubricant has soaked down the shaft for awhile, try to manually push down on the end of the shaft and work it in. That should do the trick for you unless something is broken.
You should have the T handle shifter like I do. It was used from 92-93 only. The cobra head design isn't as prone to this problem, as the button pushes directly down on the shaft, where ours has to use a sliding wedge to depress it. Let us know if this works for you.