About the best "specialty shaving" store around here is British Isles in Rice Village. It's not like some places folks talk about it, but you might enjoy popping in.
I recommend Capone's for a barber shop. It recently moved locations, and the former owner recently sold it to one of the barbers who worked there for many years, but I hear it's still excellent. I have no reason to expect it wouldn't be.
Keep in mind, though, that everything around here is just reopening. Barber shops have only been open for a couple of weeks, and non-essential, walk-in retail is just happening now. I have no idea what sort of inventory or availability problems you might run into.