Can someone with a non violent misdemeanor warrant in Travis County admit themselves into State Hospital to receive mental health care and serve their sentence there instead of jail? Or will they just be transferred to jail after they receive treatment?
You could hire a Travis County criminal defense attorney to attempt to arrange Personal Bond in advance and then doing a walk-through on the warrant. This is what former Governor Perry did when he was indicted on felony charges a few years back. It should be possible on a nonviolent misdemeanor unless you have a bad criminal history with charges like Bail Jumping or multiple Failure to Appears.
If the warrant in question is not a mental health warrant, a State hospital has no basis to admit or keep you. Call an attorney in the County where the warrant is pending.