You can try that first. I needed a little blue magic polish to get out a few small water spots. Just be careful around any etching if you have that on your blade.
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Baby oil and mineral oil are usually the same thing. I use them for razors that see use several times a week with no issues. Stainless is only stain resistant. I've got plenty of stainless and carbon steel knives. Some of the stainless will develop a patina or begin to rust rather quickly and some non-stainless ones won't (like my CPM 3V). If I get my whole razor wet or know I've gotten moisture in the pin area then I'll dry it really well and leave it out for as long as I can before I have to go to work to dry....have had some patina in areas, but no rust yet. Hope this helps.
I live in a very humid area. I run toilet paper through my scales after shaving, strop the blade and then I put a light coat of baby oil on the blade after shaving. I have yet to have a problem with any kind of rust using this method.
If you store the razor in a box, like I do, put a couple of those little bags of desiccant (the ones that come with may products and say "do not eat") to absorb the extra moisture.
What kind of razor? Pictures? A wide variety of responses, but what are you dealing with?