Are the resin beads inside old water softeners flammable?
I would highly doubt that they are flammable. Water softener beads are made from zeolites, which are aluminosilicates (minerals containing aluminium, silicon, and oxygen). They are much more similar to clay than organic resins, although they may be manufactured with a binder of some sort.  If you're really worried that they are flammable, try burning some of them - I'm guessing it would be similar to trying to burn a brick.