you are right to be concerned, however
IF the putty does contain asbestos filler, it is confined/bonded to the clay base and it is not "Friable" ( able to become to be a fine airborne dust & easily inhaled) it wont jump up & bite you, but wear gloves & dispose of it properly , also be aware that the fiberglass use to insulate the bottle compartment also presents an inhalation hazard, you should wear a dust/particulate mask & gloves when handling it. long sleeved shirt is also recommended as it is quite itchy if it gets on your skin
also please don't snort or ingest either item