You use machinery to scoop stuff up from underwater—called a dredge—to dredge up gunk or debris from the bottom of a river or lake. Metaphorically, you also dredge up old memories, the past, or objects buried in the mess in your room.
To drudge is to do hard, annoying work; and a person who does such work can also be called a “drudge.” If you find yourself saying “drudge up” about anything you’re trying to uncover you almost certainly should be using “dredge up” instead.
When you slog laboriously up a hill, you trudge up it. Trudging may be drudgery, but the act of walking a difficult path is not drudging, but trudging.
And you cooks wondering whether dredging a chicken breast with flour has anything to do with river-bottom dredging will be relieved to know it does not. The two words have completely different origins (“sprinkling” vs. “scooping”).