Mom refuses to use a walker but insists on having someone help her walk in public. Advice?

You have the right to protect your own safety. If she falls and takes you with her, you risk broken bones and injured back. So what you do is tell her (short, declarative sentence) that in the future, she must use a walker if you take her out or there's no going out. You let her know that you will not allow her imprudent choices to put you at risk of injury. She won't like it but you have to stick with it. You should also really assess whether it's time to bring in a transport wheelchair instead of a walker for longer distances. Again, you don't take her out without the chair.

My DW will take my hand or arm when walking. She really doesn't need a walker and doesn't use it. She does wander and will get tired easily. I have solved our problem by putting her into a wheel chair and even a special shopping cart at the grocery store. She really likes them and can even take a nap while I try to shop.
Other shoppers have admired this arrangement. It even helps us with most of the employees helping us when needed. This even helps me relax while shopping, which I need.
I no longer have fear of her falling, wearing me out, getting lost, or causing me to have to hurry or miss thing we really need.
Try talking to her into riding in a chair or a motorized shopping cart. Heck she has earned the privilege of riding and the respect of some others by doing this.

Does your mom have dementia?

Have her doctor order a physical therapy evaluation to determine her need for a walker. And to show her how to use it.

Then just say "no".

Stop helping her, if she doesn't need a walker she doesn't need an arm.

When she asks for assistance push her the walker, if she says no then she figures it out. Sorry mom, it is to hard on me physically to help support you, if you need any support it needs to be the walker. Repeat, repeat, repeat, repeat and repeat again until she uses the walker.

My mom helped her MIL that didn't need a walker and ended up tearing her rotator cuff and had to have 2 surgeries. She still has problems after 20 years.