Congratulations! If you were my patient, I would want to do an ultrasound to rule out gall stones. If you do have gall stones, the symptoms will often be worse after eating, especially a fatty meal. The gall bladder is located underneath the liver edge on the right side, and if it is blocked, the painfrom it stretching can radiate through to the back.
Another possibility is a kidney stone on the right side. The area you are describing is right where the kidney sits as well. An ultrasound would rule out kidney stones too.
If your gall bladder and kidney looked OK on ultrasound, I would then probably suspect some kind of musculo-skeletal pain. It sounds like your doctor is on the ball with ruling out contractions/preterm labor when you have the pain.
If it is musculoskeletal ( a pulled muscle from moving?) you are on the right track with Tylenol and rest. Heat is OK too.
Hope this is helpful!