This seems too obvious. If you don't want to stay in Texas, go to Georgetown.
I don't think any medical school teaches students how to do abortions. The medical school is affiliated with the Jesuit tradition but you don't have to be a devout Catholic to go.
I would be more concerned about your access to women's health resources and care. At least at Georgetown, you can visit your member of Congress and get involved in advocacy much easier if it's important to you.
Also, yes DC can get cold. It also can get insanely hot in the summer. Almost like Houston. Without the hurricane threats.