Young patients (under the age of 55 years) with full thickness rotator cuff tears should be referred early. In this group, the pain from a rotator cuff tear is unlikely to settle with physiotherapy alone. Otherwise, it is reasonable to persist with physiotherapy and one or two injections for 3-4 months to see if the pain will settle.
Any surgical or invasive procedure carries risks. Before proceeding, you should seek a second opinion from an appropriately qualified health practitioner.