Since the state does not recognize legal separations, can I buy a house without my mate?

I'm not entirely sure I understand your question, but I gather that you and your spouse are separated and living in Texas. In the event that you purchase a home in your sole name, that property will be considered community property because you bought it while being married. If you'd like to buy a house on your own and have it be considered your sole and separate property, you need to get a divorce from your spouse and then buy the home.