My inital try to use a wet mop left an undesirable result. Lots of residue remaining despite wringing out the mop frequently. Using a wet cloth towel and scrubbing improved things somewhat but that is backbreaking for me and still leaves patches of dust. Any suggestions, products that would make things easier?

I will probably be having some drywall sanded again and so any future tips would be helpful. Thanks

Our carpet cleaning machine has a hard floor attachment. If you do not have one, can you rent something like that, and maybe use warm water with a gentle detergent? I'm not a tile expert. Be careful if your grout isn't sealed you do not damage the grout.

Once you get it cleaned, would sealing the grout make cleaning easier the next time?

This article says drywall dust is water soluble, which makes sense.

https://homeguides.sfgate.com/...out-grout-80727.html

Good luck to you with this. I hope you find something that works.

tape a barrier (thick plastic or a tarp) down before you do any drywall work

and depending on what is being done, a damp sponge will clean up drywall seams and nail covers a lot cleaner than sandpaper

Similar problem...
When I had to clean the grout haze off a tile floor with a vinegar wash I got 2 helpers and a couple of big sponges and some buckets. I first mopped with the vinegar wash and let it set a few minuets.

Then, I rinsed with water and sponges, while I wiped with one sponge... helper 1 (wife) was rinsing out a second sponge. Just one swipe or two with each side of the sponge then hand off sponges for a clean one. 2nd helper (daughter) dumping and refilling a rinse bucket often. It went amazingly quick and worked perfectly.

Or you might just need mop a couple more times.

Next time research Youtube for wet sanding sheetrock... or cover the floor?

Don't use water. Sweep, then vacuum. Water plus drywall dust makes a white coating on stuff and is a super big pain to get out of hardwoods. Only mop after getting up everything humanly possible with broom and vacuum. Buy a crevasse attachment if you don't already have one.

Sticky lint rollers have a lot of uses. They might work to remove said dust from tile.