Jacking concrete slab with shed
If you “jack”...you crack. 

That slab weighs a minimum of 5,000 lbs. (4” thick) and will crack when you start jacking it up a few inches. 

Little known fact, concrete only has a tensile strength of 55 lbs. per square inch (as compared to compressive strength of 2500 psi ). When one edge is lifted, the top of the slab is in compression and the bottom is in tension THROUGHOUT the slab. 

No amount of levers (regardless of fulcrum points) will keep the slab from cracking. In fact, where the lever is placed could accelerate the cracking process. If you’ve ever seen a contractor remove a sidewalk, you’ll know what I mean. They use an excavator to pick up an edge so it will break and they can load reasonably sized pieces in a dump truck. 

Now, if the slab is reinforced and the reinforcing is properly placed, it could hold...but what are the odds that old shed slabs are reinforced.  

In addition, even if you were to lift the slab, how are you going to EVENLY place gravel under the slab and then COMPACT it? Impossible. It would tip and put a stress point on the slab...causing the slab to crack. 

I like the idea of trying to control the drainage around the shed and if that doesn’t work, I like the idea of removing the wood portion, using the existing slab as a footing, installing a higher slab and then reinstalling the wood shed.