Tarp it.  Put the tarp up first. set the hay in the tarp and warp it up.  Bottom, Sides, Top all need to be covered.  Place all the folds in the tarp so the face away from driven wind as you drive and so that gravity will  push water inside.

I don't know where you are traveling/camping.  But if you come from the East to the West. All public land in most western states will require Certified Weed Free Hay. I understand that can be difficult to find back East and puts travelers in the position of having to buy hay from unknown sources along their travel route.

In our part of the country, our bales are commonly rectangular, weighing ~60#-70#s each. The larger contractor's garbage bags available at the big box stores, will completely hold one bale, and can be wrapped or taped to seal the bale from the environment.

Some times handling one bale at a time, is easier than trying to wrap and unwrap an entire load each time you need one bale. Having each one wrapped individually, and having a tarp covering the whole lot, would ensure the best weather protection.

In my experience, The tarp covering the poly bags is essential.  The wind on a long trip (400 miles+) will shred the poly bags.

I use zippered 600 denier nylon bags. I have found that Justin makes the best and most durable bags. Horse.com hay bale bags are ok, but don't last as long becuase the bags only have one zipper instead of the two that Justin bale bags have. So if you break a zipper, the horse.com bags become useless unless you can find a heavy duty sewing machine to repair. I can buy them locally for $20 a bag.

It really is up to the descretion of the particular forest service or blm office how they administer the weed free program. I know that when I go into Yellowstone. They allow NO HAY into the park.  It doesn't matter if its certified or not. They also require that the trailer be swept clean. No hay or manure on the floor. They are not only concerned about weed seeds, But they don't want any Non-Native seeds to enter the park. A lot of the BLM in central Utah, not only wants Certified Weed Free hay, but they will not allow any Oat hay at all. Other places just want to see the tags on the bales, and they ask if you have been feeding the Certified hay for 3 days prior to your trip.  For us locals we kinda learn what each district is looking out for and deal with it.  For you travelers, I know it can be tough, you think you have eveything covered and then they confiscate your hay.

Pellets are always a safe choice. As the grinding process to make the pellets, destroys any seeds. that's not the case with hay cubes. They are chopped and cubed, So there still could be seeds in the cubes. So certified hay cubes will satisfy some agencies, But not Yellowstone for example where they don't want ANY seeds.

If you come out west, ( and the original poster didn't specify where her camping would be) check with the land managers for that area and see what they wil or won't allow.