Critters under the shed are certainly a problem. In my yard, we had wild cats and kittens under it. So when I decided to get a bigger shed, I did some planning.
I know you already have the shed in place, but for the benefit of others, I'll mention what I did. I opted not to have a concrete foundation for my 12' x 16' shed because the only way to get concrete back there was either with a pumper-truck (way too costly) or by mixing it in a portable barrel on my own (not worth the hastles because I'm not a masochist). So, I dug down about 4" at the lowest point where the shed was going to be, and leveled out the ground from there. I made the 'box' 2 feet wider and longer than the dimensions of the shed. Then, I lined the 'bed' with filter fabric. I dug an extra 2-3" on the perimiter into which I put some gravel and my pressure-treated border (2 rows of border high). The border is about 4" above ground. Then, I used my Dad's lawn tractor and trailer to haul 7-1/2 tons of gravel to the pit, and raked it in 2" at a time while tamping each layer. There was 4-6" of tamped gravel now in the box. After a final tamping of the top 2" of gravel, we set the shed on it.
Now, the shed itself sits on 4 x 4s but it's on gravel; not dirt. (Hopefully, your shed is not on the ground!) The critters STILL liked it under the shed! So, I took chicken wire and double-layed it so it's a tight-looking mesh, and tacked it on the base of the shed and to the box itself. Now, only the smallest of critters like field mice and the like can get under there. I've seen spots where the big critters have tried to pull back the mesh too but with no luck.
What should you do??? You state that your shed is partially on the ground. What I'd do if possible is get the shed off the ground completely. I don't really suggest you do what you want because if you're gonna do any digging for concrete, you might as well empty the shed and temporarily move it to either put it on concrete or stone. Get yourself a book from the book store where there's talk about how to build a concrete slab, preferably, for a shed. For a slab, make it about 6" wider than the outside dimensions of the shed. You should dig out the ground so that the top of the concrete is an inch or 2 above ground. YOu can move the shed if you can get ahold of a few steel pipes that are 8'+ long and roll it away, or you can drag it.
I guess you could also get some wire, dig down a foot or so, bury the wire, and attach the upper portion to the shed. This way, they can't pull it down, and when they dig, they keep hitting the wire. (The deeper the wire the better.) My best to ya and hope this helps.