I don't carry shoulder bags as I fractured both shoulders in riding accidents and it hurts to wear a bag now; only hand or elbow crook works. So, I've had to do it with all 3 of my boys. My eldest has Special Needs and didn't walk until almost 3; I have AMAZING arms, lol from all the carrying I've done.
Bottom line, you can do it. You just have to build up the arm strength. It shocks me how much I can actually carry with one arm; my 20 month old, my Tivoli, a diaper bag and a gallon of milk- no problem! HTH