I've attempted to use Minwax on a Sauder desk to re-finish it, and it doesn't seem like it is actually sticking to it after over a full day's worth of drying. When I take a towel and rub it, it comes right off.

1) How can I fully remove this so that I'm not constantly getting wood finish stains on other things?

2) What can I do to fix the areas that should be re-finished?

Did you strip the old finish(sealer) off or just sand it lightly? Are you sure the top is wood? Use mineral spirits to remove the Minwax. Minwax will not penetrate old sealer and takes forever to dry. You must remove all the old sealer before using minwax. Minwax can states "remove....existing finish"