On the surface, this is a simple moment: a man walks his dog beside a frozen river. But the title, "Preoccupied," tells us where he really is. His body is outdoors, while his mind is stuck on work. The repetition of "full" sets up the contrast. His mind is full of work, and the river is full of ice, so both seem clogged and slow, with nothing flowing freely.
The last two lines are the most puzzling. "Dark lodgings" and "a board with my name on it" could suggest a place where he stays, or a sign or noticeboard linked to his job. The poem doesn't say, so we have to guess. Either way, the speaker's name appears on a board, which hints at being fixed in a role or routine. The mood is quiet, cold and a little weary. He is surrounded by winter scenery but hardly seems to notice it, which is exactly what being preoccupied means.