I am with FHS on this. Sneakers are not necessarily good shoes to on your feet for hours and hours. And a compression can certainly result in what you describe. I know that you fear the emergence of neuropathy, but it is more likely a compression problem.
Visiting a podiatrist might be appropriate, There may be some condition causing this numbness, or there may be aspects of your feet that make you prone to this. They might recommend an insert that could help, many sell shoes, but I doubt you have an interest in those. I've also heard many recommendations on running shoes. Running is perhaps even more stressful on your feet. And things like excessive pronation are thought to lead to compression problems and numbness just like you describe.
The very best running shoe stores will have extremely qualified staff and the right support to identify key issues (like pronation) with your feet and help you select proper fitted shoes. Even if you don't buy shoes at the running store, it can help you know which shoes are best. Then you can go to DSW.