First, get your screen problems figured out. It sounds like you're underexposed. Second, get your white technique down to a point you are satisfied.
If they are just one color, registration is simple, so you don't really need to clean the screen out after every use, just keep enough ink in the well for a few prints. Keep the screens put away on a shelf, and print to order.
The only thing is that you'll need a screen for every design this way. But if you're only printing the 10 designs per season, 10 screens isn't that many. Just reclaim them for the next season's designs when that time comes.
If they are just one color, registration is simple, so you don't really need to clean the screen out after every use, just keep enough ink in the well for a few prints. Keep the screens put away on a shelf, and print to order.
The only thing is that you'll need a screen for every design this way. But if you're only printing the 10 designs per season, 10 screens isn't that many. Just reclaim them for the next season's designs when that time comes.