The problem could be the HollyTone in the Fall. It is best to fertilize just before bloom time or during bloom time. The Fall is exactly the wrong time. In fact Summer and Fall are both bad since it can keep the plant from going dormant. It can force new growth on the ends of branches that is tender and freezes and turns brown. HollyTone is a very good azalea fertilizer but use it at half the rate on the package and only use it once in the spring around bloom time.

If just some ends of branches die, then it could be borers. Ends of branches die when a Rhododendron Stem Borer, Oberea myops, Dogwood Twig Borer, Oberea tripunctator, or Rhododendron Borer, Synanthedon rhododendri, is in a branch. Borers only affect the portion of the plant away from the roots from the borer. If the borer is in the main trunk, then the entire plant will wilt and die. The plant can be save by cutting off the area with the borer and letting the plant regenerate from the roots. There are no conventional insecticides that will kill stem borer larvae once they are inside the branches. The best control option for homeowners with only a few plants is to prune out and destroy wilting branches in early spring or late summer.

Here is a link that might be useful: How to care for rhododendrons and azaleas.