I'm sorry to hear of your condition. Temporal arteritis can be very difficult to manage. However, prednisone usually does not cause tearing. It can certainly cause swelling, including around the face.
There could be clogged tear drainage ducts that is causing your tearing. An ophthalmologist can test this very easily in the office for you.
As far as the prednisone, sometimes we can change to another medication such as Methotrexate or Imuran or Plaquenil to treat this in order to spare the steroids.
Hope this helps and thank you.