Half the dust just goes into the air and resettles elsewhere. It seems like I am making the situation worse.
How can I wipe the dust in a way so as to avoid just pushing it around so much?
I did some web searches but none of them seem to have much helpful advice on this.
Dust generally falls and settles. Dust raised surfaces with a duster, most of it will settle on the floor. Vacuum floor, then mop.
Yes, not all dust will be collected in this way but the vast majority will.
Note that I don’t use a damp whatever until the last step. Dust is so much easier to deal with when it’s dry. When it’s wet you’re just pushing it around.