A lot of sheds I've seen don't have wood lapping to the floor.
Definitely your problem is that it isn't raised off the main ground level, i.e. your patio is the ground level and water will potentially pool underneath shed, certainly not drain quickly away from the edges.
As I said before, my shed is raised above ground level with patio slabs positioned where the joists are. Not a sign of damp.
I'm not sure on the easiest solution but perhaps a couple of palette trolly jacks could lift it up enough? Are you able to get behind the shed? (climbing over it of course).
Will plastic alone resolve your problem? Probably make it better but it wouldn't be the solution I would go for. My last shed was only on patio slabs and didn't have any damp issue and sat there for 10 years. When I took the shed down and removed the slabs, you could see the water had channel underneath washed away a lot of the fine hardcore I'd used - just shows the slabs did the job!
If you could get 6-10 patio slabs underneath (one side at a time) that would probably be easier than trying to get plastic underneath (probably requiring the whole shed to be raised off the ground).
Just my 2p's worth