From years of business travel and "unscheduled" extensions to business trips I learned a couple of tricks in doing sink wash.
Once you have washed your socks, underwear, undershirt or other stuff that doesn't matter too much if it wrinkles, wring them out as much as possible and then lay them flat on a dry bath towel. Don't overload the towel. Just a couple of pair of socks or one t-shirt per towel. Roll the towel up with the garment inside. Gently pat the towel to absorb the water from the garment. Do this for a two or three minutes.
Unroll the towel, remove the garment and then put it on or over a hanger and hang as close to a vent as possible. Everything should be dry in just a couple of hours. If an iron is available you can iron it if you want wrinkle free underwear.