Texas doesn’t have to cut funding to sanctuary cities. Counties and cities in Texas are wholly subservient units of the state gov’t.
The state can, by decree, override the ordinances and laws of any city in Texas.
The real problem, of course, is Republicans controls 2/3rds of the Texas House, which is still headed by Speaker Joe Straus, first elected Speaker by 8 Republicans and 142 Democrats.
Speaker Straus has refused, session after session, to bring up sanctuary cities, just as he does with pro-life legislation.
The real power of the Texas Governor is to call special sessions of the legislature for specific issues. This has been how pro-life legislation gets passed in Texas and while Straus is Speaker, the only way sanctuary city legislation will move forward.
8 GOP and 68 Dems elected Staus. 76 is majority, 150 total.