You definitely do.
I second everything Des said.
I also strongly recommend going to a running store where they’ll let you run around in them AND WATCH YOU RUN. Do some research and find a store with real runners who can help you decide if you need stability, cushioning, if you pronate… etc.
If you get shoes for over-pronators and you don’t actually run that way, your shoes will force you into poor form.
Once you get that advice, you can, of course, order the same shoe cheaply online. I usually buy my first pair from the helpful store… I figure it’s help tax.
Try not to buy them on looks. I have, several times, ended up with the ugliest but most incredibly comfortable and marvelous shoes. I would run ugly any day to avoid injury.