Dallas as a whole is for sure not a liberal enclave like Austin, however, the downtown/uptown/deep ellum bubbles of Dallas most certainly are. If you are coming from SF/CA, Colorado, Austin even, and you are in that part of Dallas, you will feel at home. The "Texas" stereotypes you have for sure heard about are pretty absent here.

For affordable housing, you can definitely get it, but if you want something close to downtown/uptown, it's a good step up. Also, the housing market here is ridiculous, one of the fastest growing markets in the nation. You will be paying above ask in a bidding war for anything you find. And you better be ready to put your money down, no contingency, sometimes before you get to even walk in the front door to see the house. It's that insane.

Which city sounds better for raising a family ?

I’ve lived in Texas my entire life. Austin is better for raising a family than Dallas.

If you considered the suburbs of Dallas though, I'd say they are about the same. Frisco, Plano, McKinney, etc.