I guess it's preference but leaving a unrinsed blade in my razor would be like putting on a dirty pair of socks. Why. It just takes a second to remove and rinse the blade and razor. I do it for the same reason I wipe off my Craftman tools after each use. Just because...
With my vintage Gillette razors and modern SS blades I'm in the leave the blade in the razor, rinse it in hot water, pat it dry and leave it a little open to dry camp. I do only wear clean socks even if since moving to coastal South Carolina I can't remember exactly when I last wore socks. Oh and my tools all do get wiped down with WD-40 on a rag after each use but that's because they stay in a hot and humid garage and I don't exactly use them every day.
I rinse my razor under the tap after each use. If it is a TTO, I'll open the doors while it is being rinsed. I shake off the excess water and leave it out on a towel to air dry. I swap out my DE blades after 3 shaves and my SE blades after 7 shaves. I prefer not to handle the blade any more than necessary. This works OK for me, but it is interesting to see what everyone else does. The brush also gets rinsed out, dried as best I can, and also left out to air dry. My dirty socks thing about shaving is filling the sink with water and continuing to use it during the shave. Instead, I run a small stream of water from the tap and use fresh water while I'm shaving.
I can get them here still locally, do you need some? If so, shoot me a PM with your address and I'll get them out to you asap. Also, have you been across the street? All things SE, may be more help there as well. Just let me know about the blades.
After every shave; I take out my blade lay it on a cloth. Using a soft tooth brush clean my razor(s) under water. I have a wash cloth I lay the razors I am using on so not to scratch any thing. With a dry cloth I wipe dry the razor(s). I then clean the blade under running water, pat blade dry, then I put it all back together again with the handle loose or the doors not tight. If the blade has 2 shaves on it, I toss the blade and put the razor on the hanger whithout a blade in it. Every time.
My razors are cleaned after each use. Open or take apart razor, rinse razor and blade in hot water, pat dry blade, wipe down razor dry. replace blade, reassemble or close razor. If blade was used 3rd time it's replaced.
I keep the razor blade in the razor, i have seen a few people cut themselves pretty good handling a blade more then they need to. I dont keep a bunch of razors in the bathroom, usually one DE and one SE. They get cleaned once in awhile but im not ocd about it, except if im using an adjustable in which case ill soak them for a few minutes once a week in barbacide to keep all of the moving parts from getting gunked up. I hate waking up lol, i dont have the patience to be cleaning and polishing razors like some do. I shower first thing after waking up, my thought are more focused on hoping the apocolypse or some other event will occur and give me a reason not to go to work.
I have three little ones so honestly I keep my blades kept high In a closet and my razor I rinse it and the blade off quick dry and place it back in the razor and place it on top shelf of med cabinet. seen them climb to much so I never leave a razor out in the open.
Once a blade is loaded in the razor, it stays there for 3 shaves. About 6 days or so. Never had any rust issues with any of my razors, but I always place my razors under a ceiling fan after shaving !!!
I've left them in and taken them out both over the last few years. I'd have to say I've never seen any corrosion issues with any of my razors or blades but I am a believer that you can get a build up of crud on the edge of your blade that will inhibit it's effectiveness. Do what works best for you according to your lifestyle would be my advice. I've also never tried any carbon steel blades which depending on what alloy they were made from might be a very different story.
I open the TTO doors or loosen the three-piece and rinse with hot water. I leave the blade in however.