I have had a "problem" many times were I forget to wash towels. I take a shower and BOOM! I have nothing to dry off with. The worst part is I shower in the bathroom closest to the busiest part of my house, so my family sometimes sees me with nothing to cover my skin. I do not like to use the blow dryer because I am trying to go with a blow dryer free lifestyle. Also, the hand towels I can use as a substitute are on the opposite side off my house on the bottom floor so I can not get to them. I also shower at night, so I have to sleep really wet. I try my best to dry off with my hands, but my hands are small and do not get rid of a lot of water. I need a new way to dry off, also note I DO NOT HAVE CLEAN TOWELS!!!!!
Assuming that you didn't go to the bathroom naked, you will have your worn clothes with you.
Unless they are really dirty from sweat, dust or other outdoor dirt, they will, in a pinch, be a suitable towel stand-in. If you wear layers, e.g. a sweater over a t-shirt, the inside of the sweater and the outside of the t-shirt will be basically clean because they neither touched your skin nor gathered dirt from your surroundings. For other clothing styles, the same principle applies.
Extra hint: T-shirts are very good for soaking up moisture from your hair.
First, watch this TED video (3:41).
Then start shaking off your hands and other limbs (note: jumping jacks might not be a great idea if the surface is slippery), and then brush off the water from your skin (e.g. arms, shoulders, everything smooth) as if you were brushing off a bug or a few crumbs, but it's brushing off the water droplets instead. The size of your hands doesn't matter that much in being able to move them quickly over the rest of you to remove most of the water.
You said the bathroom is near the busiest part of your house, then you can call someone and tell him/her to give you a towel. Or if you insist on not using a towel then ask for something else - t-shirt or other clothes.
Or if you have taken clothes with you in the bathroom (clean or old) use them to dry your skin and after that you can change them with clean clothes again.
Or you can take big piece of paper (usually every bathroom has toilet paper) and with tapping movement. No rubbing because the toilet paper will become on pieces.