How can I protect a shed floor from ground moisture?

Your shed is in direct contact with the ground, and the joists/floor are made of wood. If that remains the same, then your floor is going to rot. OSB is especially poor for such close proximity to the ground; at the very least you want plywood which won't warp permanently like OSB does when it dries out.

If you want a shed floor/subfloor that is wood and lasts more than a few years before rotting, then you have two proper options as I see it, and neither of them is going to be a quick fix:

1. Dismantle the shed, lift up the subfloor frame, and put 8x8x16 concrete or cinder blocks underneath each corner and under the center of each of the floor sections in your photos. You can probably keep the roof and each wall in one piece if the shed was framed well and you have some help.

   Sometimes you can also have a company bring in heavy machinery to lift up the shed, but that depends on the ability to brace the framing and fit the machinery in place, and it's hard to tell from your photos if you have enough space for that.

2. Excavate out the ground between each of the floor joists (and underneath most of them), leaving columns of dirt/soil underneath the corners and center, and then put bricks, concrete/cinder blocks, or pour concrete forms around those dirt/soil "columns". This will achieve a similar (but lesser) outcome as option 1 above.

As you can see from these two options, the main thing you need that your shed is missing is airflow (or an impermeable material of a certain distance/thickness). There's a reason building code requires a certain distance between wooden floor joists and the ground, and that reason is the problem you're experiencing.

If the two options above aren't palatable, you could try digging out just a little bit and then putting in a sump pump underneath the floor, and/or putting in a waterproof membrane in the cavities like people get in their crawlspaces to help water abatement, but those are going to be just as annoying to do as the above 2 proper options, in my humble opinion, and won't be nearly as effective.

If the shed is properly spaced above the ground (check your local building code for the minimum height above exposed ground for floor joists in uninhabited structures), you shouldn't need any special sealant or wrap material for the floor to have plywood or spruce plank floor (as you mention) which lasts 15+ years even in a humid region.

If you say the shed weighs 1 ton is this full or empty? Your best bet would be to figure out how to raise it off the floor and set it on pad stones.

I've built several such sheds in a climate that's quite moist and humid in summer. My strategy is to frame the floor with treated lumber, as you've done, then overlay either poly sheeting or house wrap, stapling it to temporarily hold position. You'll need to add some lumber around the perimeter to carry the floor along the walls, so do that first.

For flooring I've used untreated SPF tongue-and-groove plywood, which I give two coats of quality urethane after installation. This is mostly to protect against moisture draining from snow equipment and lawn hoses. I would not use OSB since it's more susceptible to moisture swelling and delamination in wet scenarios. OSB would be fine for the walls.

The first shed lasted 15 years with no serious degradation of the floor. The second one is about 3 years old and doing well. If you wanted to go the extra mile, sealing all sides of the flooring boards before installation would probably do some good in the second and third decades of the shed's life.