There are many types of materials at your hardware store that you could use as spacers if that is the way you want to go. All would probably be a little frustrating to varying degrees whenever you want to remove the screens for cleaning. Maybe just a few small stainless steel screws screwed into the side of the frame that could be adjusted for depth might suffice. Alternatively, have some new screens fabricated. Options include a couple of leaf springs in the side frame to keep the screen snug.

If you haven't already resolved this problem:

Coated deck screws likely would be less expensive than stainless steel but that wouldn't solve the problem with the spaces between screen and frame. You might consider some weather stripping for that.