It's 12x22, and yes, I've jacked and leveled many sheds/garage over the yrs. This one is set in a spot that is all gravel and drains well. Once this one is set on blocks, it won't move.Have you jacked many other buildings up to level them?
If so, maybe you know things we don't. If not, you might reconsider that plan.
Two observations on it. It is a lot harder to jack up a building than to jack up a car (the only other big thing we have jacked up.) And it is better for it to settle straight.
Ours didn't settle straight. Or wasn't built straight. Or both.
Our foundation was perfectly level. The building wasn't. It was built elsewhere and hauled to our lot where it sat on unlevel ground for a few weeks before he came back to move it on the foundation. The builder got the doors to work by jacking one end up and putting shims under one corner. That made the floor bounce, the blackjack 57 over the floor to crack, some nails in the floor to work out, the windows to not fit, etc.
We jacked it up enough to get the shim out which fixed the floor problems. Then the doors couldn't close. I'd never jacked up a building before. We had a jack designed for similar jobs - not a car jack. I still never want to do that again... trying to get the jack under the building was awful, working the jack under that much weight was awful, reaching under the building to get the shim out was scary, letting it back down was awful.
I think much of the problem is everything built (the frame, the wall sheathing, the roof deck, the roofing, the frame for the roosts, ect) braced the lean and rack of the building. It did NOT want to adjust.
It took a LOT of tries with come-a-longs and braces, to rack the front of the building straight enough for the door to close if the weather is dry. It tended to settle back despite the braces. And far too much shaving of the door and the door frame to get it to close in damp weather. It also popped a couple of the wall boards off the frame (our has board and batten walls), and cracked a couple of other boards.
We left the other end of the building leaning.
It looks like your building is about 10x6? That is about half the size of ours. Hopefully that is enough that yours isn't as awful to jack up. I don't think being smaller will help much with the other problems though.