I used to go to this place pre-pandemic up in north Austin and they do. I just started shaving my own head so I no longer need a barber but they were great. They appear to have more south locations: https://finleysbarbershop.com
South side is lacking in good barber shops as far as I have seen. I go to Little Barber near the university and they've never done me wrong. Their online booking is a standard experience.