Bronze disease is unlikely. You have common malachite copper corrosion products. It appears from the images (and this is only based on the visual) that at least the “star” badge, and possibly both. are brass plated iron objects. The sherrif’s badge will have some surface damage /plating loss once cleaned. The star appears to have lost much of its plating as the substrate iron base has corroded through the brass.
The sherrif’s badge can be cleaned (by a conservator) with a very mild metal cleaner (no sulphides or ammonia content). The star will require more conservation attention but will not be “restored”.
Store both in a neutral container in a climate controlled area. If necessary follow the previous suggestions regarding silica gel. Coating the objects after cleaning with a microcrystalline wax of a lacquer may slow down the reaction, but if stored properly this should not be necessary.