If the shed isn’t off the ground, then moisture is going to seep up from the ground (even if it isn’t directly sat in puddles). This is where your floor problem is going to be coming from. The only way to solve it is by putting the shed on a raft to lift it off the ground by a couple of inches and so air can flow underneath. Probably not an easy thing to do with an already constructed shed and if it’s old I may not withstand the movement of doing so (even if if is possible) and it may be that a new shed may already be needed if the floor is getting wet, it is going to rot out.
Ie my shed is on a concrete base which sits higher than ground level so water doesn’t sit on the concrete. The shed itself then sits on a slats which lifts it a further inch or so off the concrete base so my shed is about three to four inches higher than ground level so doesn’t sit in water
Once the shed is wet inside on the floor, without sufficient ventilation it is going to just be stay damp inside as it can’t escape sufficiently and of course a damp UK winter it’s unlikely to dry out sufficiently anyway, which in turn obviously means everything inside is going to be exposed to excess damp.
Any holes which are allowing water in can easily be filled, boarded over but if there isn’t an actual vent in it, then the holes may be allowing for some ventilation unless and until water actually manages to seep in through those holes. The biggest problem sounds to be if the shed is directly on the ground though.
Are you able to move piggy indoors to be out of the damp until you can decide what to do about the shed? It won’t be healthy for him/her to be in a damp space
