The best way to prevent the two badges getting worse it to keep them in a polythene box with a good seal, with a polythene bag of silica gel beads (prick small holes in the polythene bag to allow air interchange). Aim to have about 1/10 of the volume of the box filled with the silica gel. Use indicating (colored) silica gel so you can see when it needs changing, to maintain as low a relative humidity level as possible. Make sure the badges are padded with Plastazote (polythene foam) or acid-free tissue so they cannot move around and physically damage each other.
Do not keep the leather bag with the badges as this may well be making the corrosion worse – it will have kept the humidity around the leather higher and may be exuding fats, acidic vapors and other chemicals which encourage corrosion. The leather itself would suffer if kept in the very dry environment which will help preserve the badges. Keep the bag in separate packaging but please make sure the association between the bag and its badge is kept – record them together in photographs, make sure their records state the association clearly and cross-reference their numbers if they don’t share the same number.
A conservator with suitable training and experience would be able to minimise the corrosion, clean the surfaces to remove residues which might be causing problems, stabilise the copper alloys present with benzotriazole and coat the badges with a protective layer. As the previous poster has mentioned, the situation is complicated by the possible plating and lacquering, which will influence the types of treatments and materials which can be used. The American Institute for Conservation AIC can help you find a suitable conservator.