I rinse the blade under the tap during the shave, toe down, pivot high, so no water gets in the pivot or scales. After the final pass, I rinse the blade under hot water, wipe it on the towel I used after the shower, and leave it open while I clean up and get dressed. Then, I put the razor away. No issues with water spots or rust. Easy Peasy.
When I shave super early in the morning for work and am trying to be as quiet as possible because everyone one else in the house is asleep (as well as everyone else in the neighborhood), I don't rinse, but wipe the lather on a towel. When I'm done shaving, I give the blade a final wipe on the towel and put it away. I also wipe my face with the towel, no rinse. I've learned a good cream also acts as a moisturizer, for me.