Concrete. We poured a new floor in the garage and the excess was wheelbarrowed out back into the shed. About two inches or a little more would be good. You can get some bags of concrete (Sakcrete), mix in your wheelbarrow and dump into place. Ours was just dumped and rough leveled, no troweling, and makes a very clean floor.

It sounds like you may have water issue. As a result, I would try to get the shed up off of the ground to prevent future issues.

There is a post for free shed plans. Look how those builders created floors.

Water is the problem. Let water vapor move or completely stop it.

Concrete stops most of it. The gaps in the shed will allow the dampness to evaporate.

Gravel is also a good shed surface - it is not too objectionable to walk on or park a tractor on, and it allows moisture in and out.

Or, you can put skids down which are lengths of 4x4, attach plywood to the skids, then lift the shed onto that. The skids will provide air circulation and slow mold growth.

Concrete is the way to go, I'd try to elevate the shed, keeping it a dry as possible.
The wood should not touch the soil, a foot higher may suffice. And only pressure treated wood contacting the concrete.

A tall shed can look really nice