I'm part of a Walking group - actually it's a Marathon training group and we have a Walker's division. There are a lot of newbies that join every year and we have running shoe specialists who come and advise us on what to get.
Asics - is a great brand I particularly like the Kayano, .
Saucony - is also teriffic but I have a narrow foot and heal and I kept sliding so it didn't work for me.
Adidas - actually has a brand with great support for Walkers I wish I could remember what it's called but ask for the new one for Walkers.
And some of the Walkers like the Brooks brand, I can't remember specifically which one.
Also, remember your foot needs support for all that ankle action and pushing off. They should also be asking you to show them which way your foot leans. Mine lean in and that's called pronating.
Hope that helps.