I put plastic down first to seal out moisture, then pallets on top for airflow. I was going to use a tarp instead of plastic but decided against it as most are only “water-resistant” and would eventually be permeated.
I also did way too much reading on stacking the bales. I stack the first row on its side, perpendicular to the pallet slats then the second row is sitting normally (strings facing up) perpendicular to the first row. Then repeat going up alternating the orientation of the bale (on its side or strings up) and alternating direction (either perpendicular or parallel to the pallet slats). I also leave a couple inches between bales for increased airflow, but this can begin to take up a good bit of space and may not work if you get larger loads of hay.
To keep dust down, I sweep excess hay out and if ever gets really bad I suppose I’ll break out the leaf blower. As far as preventing the back corner bales from getting dusty like you mention, alternating which end of the shed you are getting hay from (assuming your ShelterLogic will have two doors) will eliminate the untouchable corner hay issue.
Edited to add: Just saw your post about flood concerns. I should’ve mentioned that I extended the plastic up the sides of the wall a few inches and taped it so water wouldn’t get on the plastic and leak in.