I do a pinch wipe type of thing with a towel when sharpening but pay special attention to contact the edge with as little touching as I can while still getting the job done.
When I am done shaving I put a little dab of hand soap from one of those pump bottles, on my finger and thumb and do a couple of careful pinch wipes of the edge to clean off the shaving soap scum.
All the shaving soaps I have used tend to leave a soap scum in the sink, so I assume there is some on the blade as well. The handsaop in the dispensers seems to dissolve the scum easily and then I rinse with hot water. A single folded piece of TP is used to carefully dry the blade first and then the inside and finally the outside of the scales.
That is probably overkill but I don't want a damp soap scum layer residing on the blade until it is stropped the next day. Also I don't want it on my strop, even though many advocate using lather on their strops. Maybe I will try that on an old strop someday, but for now my new strop will stay saop free.