Yes, you can continue to receive California State Disability Insurance (SDI) and Paid Family Leave (PFL) benefits even if you move out of state. These benefits are based on your previous employment and earnings in California, not your current residence.
Family employees - Services provided by (1) children under the age of 18 employed by a parent or partnership of parents only, (2) spouse employed by spouse, (3) registered domestic partner employed by registered domestic partner, and (4) parent employed by son or daughter are not subject to UI, ETT, and SDI.
No, Texas doesn't have its own state disability program. Only five states have a state program (California, Hawaii, New Jersey, New York, and Rhode Island). Residents of Texas can apply for federal disability programs (SSDI and SSI). Read more about SSDI and SSI here.
You can opt out of CASDI and create a private plan for voluntary disability insurance (known as a voluntary plan) if you meet certain requirements. A voluntary plan must: Provide all the benefits of SDI. Include at least one benefit that is better than SDI.