Ugh. I hate those plastic zippered bags! Cool at first, but they do get to be a hassle. To be honest, I just keep my show blankets stacked, out of plastic, on a shelf inside the dedicated “horse room” of my house.
At the barn, all of the saddle blankets for shows are also kept folded in half and stacked neatly on a separate shelf in a separate room (not the general tack room). There have to be 20 of them, in all different stripes and patterns. Only a couple still have their plastic zipper bags. When we show, we pick the ones we want to use and just slip them over the blanket racks in the trailer.
Just to be clear, these are wool blankets, that go on top of felt liners. There’s no fleece on the underside. So they don’t get sweaty or wet when we use them.
If you’re talking about the fleece ones with a nice blanket stitched on top, so one piece total, then I’d just make sure the underside is clean (hand wash/air dry) and store accordingly. You can buy or make inexpensive blanket racks.
(You can probably tell I’m not big on all the zippered bags prominent in the western world. Ack! There’s a bag for everything!)