Do civilian employers still have to pay me my salary while I am at annual training?

I don't think so, but I would recommend you contact your nearest Department of Labor and they should be able to answer that question. Because when I looked up the USERRA, I didn't see anything stating that salary employees are to be continually paid. I could be wrong, though. One caveat on something I read is that if you worked part of the week in your civilian job and then left, you have to be paid.

There never was a requirement to pay anybody anything. You must have been listening to urban myth. The Feds and some states like Texas pay up to 15 days of military leave, but that's an administrative choice (regulation), not law (USC). That's it. No other entity is required by law to pay you. It would fail in the courts as an unfunded mandate thing or improper taking of property. What some companies do is make up the delta in pay (time limit there too) if your MIL pay is less than regular pay. Others just call it unpaid leave and bitch about having to take you back.

It depends on your employer. If you work for a state, county, or municipal entity, then I believe Texas law provides that they must continue to pay you. Many private employers also provide paid military leave. Make sure that your employer knows that you are taking leave for military duty, not just vacation time. These folks can give you more specific guidance:
https://www.esgr.mil/About-ESGR/Contact/Local-State-Pages/Texas