First time mom – Obgyn and hospital recommendations in Houston

I found out I’m pregnant 2 weeks back and I’m in my 7th week. Based on various posts in Reddit and other forums it seems like TCH (Texas children’s) is the best hospital in the Houston area. However the earliest appointment I got here is about 4 weeks from now which is already the 11th week. Isn’t this too late for a first appointment? I can get an appointment with an obgyn who works with women’s hospital in 2 weeks.

The person who scheduled my appointment at TCH told me that they can’t see me if I visit another obgyn before my first appointment. Is this true?

How does TCH, women’s hospital and Methodist compare?

Also, how important is the hospital as compared to the obgyn?

Thank you