When the police book you in after they arrest you without a warrant can they charge you with a crime before they get a warrant in the state of Texas?

A warrant is an order for an arrest issued by the court. But many arrests do not need a warrant. If you punch someone in the face, and police are called, you will be arrested without a warrant, and one will not be needed. If say, you failed to show up in court, then a warrant for your arrest will be issued, to alert all officers that you should be arrested on sight. So the warrant itself is not the thing 'allowing' you to be arrested.