I have pain in my lower back also.
When you push a shopping trolley you are in effect leaning on it, and this means you are putting some of your weight onto the trolley and not into your back and legs.
I have a muscular problem and find that leaning on a walking frame or using crutches eases the pain.
HTH.
Flexion (bending forward) can have alot of benifits for lower back pain. Not to mention, you are taking weight off of yourself and putting it on the cart. It's a double benifit as far as pain goes. If this has been ongoing, i'd get it checked.