Best way to weatherproof OSB

There's no such thing. OSB will swell, warp, and degrade unless it's kept safe from sunlight and perfectly dry. No paint will do this. If you want to use OSB to sheathe the walls of your shed, you need to build those walls like the walls of a house: with a weather-resistant barrier like Tyvek over the OSB, and then cover that with siding of some sort. Vinyl siding is cheap and DIY-friendly.

If you want to build a shed out of a single material without needing to do any of this, I recommend concrete blocks.

Your reasons for needing a shed are exactly why you shouldn't use OSB as an external covering for your shed without something over it besides paint to protect it from the weather. Paint is not a moisture barrier. OSB grows and spreads mold exponentially faster than regular wood. If you're already having mold problems using OSB at all would be pretty bad idea.

Have you considered using something like Thompson's Water Seal? They're designed for porous materials and wick away water, but I'm not sure how they'll hold up

I know this is old but.....I have A frame chicken coops made from OSB that I stain / seal with whatever deck stain Lowes has in their discount rack. I don't let any part of the OSB actually touch the ground; treated wood runners for that. They are several years old and holding up just fine. They suck up the stain / sealer but after several coats, the water beads and runs right off. No swelling and no issues on edges.

Bit of an old thread, came across it googling. But for anyone else who does. You can seal osb boards with bitumen paint. The type that dries so its no longer wet, but still slightly soft, like firm rubber. I did this for a winter shield for my outdoor aviary & its been up 6 years so so far & still in the same condition when I first put it up.

Bitumen paint is no different to felting it for shed roofs. So long water or sunlight doesn't make direct contact, it'll be fine.

Just remember to cover your fixing screws with it after or water will find its way in via screw fixings.

If there's a will, there's always a way!

I recently built a camping shelter (12x24 wood frame platform with "roof" and "walls" with a tent inside) on some land I own. I was on a serious budget but I wanted it to be big enough to be worth while so materials suffered. I framed it out pretty good but when I got to paneling I ended up Frankensteining it.. For my (first) two short walls I went with painted OSB. I figured it'd be good enough for my purposes. Like others have said.. It swelled and started softening up within a few months.. In order to avoid replacing those walls altogether I just wrapped it in house wrap and that has protected it without issue. Looks like crap.. but this was never built with "looks" in mind.

I went with t1-11 for my front wall. It is SOOO much easier to paint t1-11 and it looks MUCH better than OSB covered in house wrap. Before my last wall, I ran out of money my wife would let me spend on my pet project.

Not exactly up to code... but I ended up doing that back wall in rigid foam insulation. with wood prices being as insane as they are now. and if you don't care about looks either.. it may be an alternative. (assuming you haven't finished your shed in the 5 years since your post. lol) It actually does a great job of keeping out water and standing up to UV. I sealed the joints and voila. Obviously you'd need to add some rigidity somehow since foam doesn't exactly tie a building together. Does a much better job than the OSB and T1-11 in retaining temperature (especially reflecting heat). I (carefully) used roofing screws with the washer to fasten to the frame. I can't drive up very close to my build site. So the negligible weight of these "panels" made them an attractive choice for me as well.

The "roof" is wood framing with more of the foam panels supporting a large tarp. I have to replace the tarp every year or so, and like I said, no one is going to accuse it of having a finished look. Nor is it sealed to keep out insects and mice.. but it's pretty much what I envisioned when I set out. Just a place to stay somewhat warm during the winter.. and keep all the junk I have out there dry. I got away with 288 sq. feet for under 800 bucks.

I stupidly built it under enormous hardwoods by a creek. So I am working on another (more solidly built) cabin at another spot free of large trees. It's just a matter of time until one of those trees drops a big branch or falls completely down on top of it and destroys it so I'm not spending much time/money/effort improving it.