What can I do to eliminate the stickiness of my oil-based wood stain?
If you were using a combined stain/varnish (likely with the Minwax brand), it was probably outdated. They can expire. Removing it's going to be essentially a matter of stripping the wood bare and starting again.

If you were using a real stain, they normally aren't applied like paint -- they're flooded onto a surface, allowed to sit a bit to soak in, and the excess is then wiped off before the piece is allowed to dry. This provides only color, not protection; a layer of varnish/lacquer/shellac should be applied over stain to seal it.