Your question indicates you already know the answer: Get a different bag that distributes weight differently, across your shoulders and hips. The sooner you find something with better ergonomics, the sooner you can learn to work with it efficiently. The longer you delay switching, the more likely an injury will eventually force you to switch.

Messenger-style bags with with canvas straps should be used for short periods of time. Longer than about 15 minutes, long enough to walk across a small campus or to enter a building from a vehicle, causes excessive musculoskeletal strain. As we age, our bodies are less able to tolerate minor damage, and "normal" activities can cause pain, as you've noticed. This is your body telling you that (potentially serious) injury is imminent and it's a bad idea to continue.

A temporizing measure you can consider is to replace the strap with one containing neoprene, which can absorb energy before it is transferred to your body. This would allow you to use the bag longer before experiencing discomfort. However, I strongly suggest you switch, sooner rather than later, to a different style of bag that is designed to be used for longer periods of time.