In the city most people use fold up grocery carts for groceries. They’re inexpensive and take up very little room since they easily fold flat. It doubles as a way to transport laundry if you don’t have machines in your apartment. I linked a very cheap one. You might want to splurge another $20—$30 and get one with really good wheels and more durable depending on your sidewalks.
A lightweight messenger bag is better than a heavy purse on one shoulder, which is what a lot of women use. I cross my purse over my body like a messenger bag to relieve my shoulder and distribute the weight better. That’s a messenger bag compared to a purse, but a backpack for heavy things will be much better. Especially since what you are carrying is heavy, and you already feel the discomfort. If you feel it, try not to use that method.
Make sure whatever you buy isn’t heavy before you even put something it. Some of the prettiest bags, attaché cases, purses, backpacks, are made from very heavy material. Forget that.
Why were you unsuccessful finding a backpack that works? Why were they inadequate?
One thing to remember is a backpack can be accessed without you noticing so you would want to make sure your money and ID aren’t easily snatched from the backpack.
Someone suggest a bag/luggage with wheels. I think that’s a great idea. You can get backpacks with wheels, but I think just go for a more regular looking rectangular piece of luggage with wheels. You can get one with a strap and handle option.