westtexas, I understand the weather issues, as I used to live in AZ. And now that I live in a 4 season climate, I don't ride once the snow is on the ground. But, a lot of people do. Everyone has different tolerance for riding in weather. But, you can commute without showers, if you want to. I did, and I also was expected to look "professional" on the job. It just takes planning. I would bring all of my clothes to work for the 2 days I commuted on Monday. I had all of my hygiene supplies at work. I took a shower before leaving and rode slower than usual (heck, it was 5:45 in the AM). When I got to work, I would check email and get my stuff out while I cooled down. Then, I would go into the rest room, wash my face and totally wet my hair, which is very short. The important parts would get washed with either a facecloth and soap, or a bath cloth, which is like a giant baby wipe and is heated in the microwave. Then I would get dressed, put gel in my hair and do my make up. I don't think anyone would say I looked unprofessional.
I also think there's much more of a car culture in the sun belt states. People tend to live far away from services and don't walk anywhere. When I first graduated from college, I lived near downtown Tempe; I used to ride my bike to do errands and walk places, despite the temperatures. I did it early in the AM or evening. And that was when no one rode bikes anywhere!