Are there effective pain killers that don't cause addiction?

Yes. There are meds that are not addictive, such as nsaid's, ex Ibuprofen or naprosyn (naproxen). There are other class of meds that are nonaddicting as well depending on the type of pain. This can include certain antidepressants or seizure medications. There are topical therapies as well as non traditional, acupuncture, physical treatments such as osteopathic or chiropractic. Injections, massage therapy, .
Yes. Unfortunately, they may be less effective and patients are not really looking just for pain relief but for the relief of emotional distress, anxiety, and depression that completes the picture of chronic pain.