Ok, I don't have advice on the drying rack but....Where in this world are you travelling to? Asia has some very good deals on laundry services.
I cleaned stuff in the sink all the time and used a towel rack or the shower rack (if available) for drying. Usually for t-shirts I would use hangers. Even in the hostels I've stayed at there are usually a couple of hangers available.
One tip is to take a towel and roll up your laundry individually and ring out as much water as you can before hanging. I found this absorbed a bit more water than just doing it by hand!