You could also try a microfiber mop flat pad set up with a recharge bucket.
Like these:
https://www.directmopsales.com/Microfiber-Twisted-Closed-Loop-Mop-Pads-s/220.htm
https://www.directmopsales.com/Microfiber-Mop-Recharging-Bucket-s/160.htm
These can add a little water to the floor to soak...