Dear Berumen96,
SB 1566 of the 85th Regular Session of the Texas Legislature is now a law located under § 37.0815 of the Texas Education Code. Here is the text of the statue:
Sec. 37.0815. TRANSPORTATION OR STORAGE OF FIREARM AND AMMUNITION BY LICENSE HOLDER IN SCHOOL PARKING AREA.
(a) A school district or open-enrollment charter school may not prohibit a person, including a school employee, who holds a license to carry a handgun under Subchapter H, Chapter 411, Government Code, from transporting or storing a handgun or other firearm or ammunition in a locked, privately owned or leased motor vehicle in a parking lot, parking garage, or other parking area provided by the district or charter school, provided that the handgun, firearm, or ammunition is not in plain view.
(b) This section does not authorize a person to possess, transport, or store a handgun, a firearm, or ammunition in violation of Section 37.125 of this code, Section 46.03 or 46.035, Penal Code, or other law.
This law permits teachers with an LTC to transport or store their firearm in in their vehicle as long as it remains out of plain view. It does not require that the firearm be unloaded. It can be stored – loaded or unloaded, chambered or unchambered—as long as it is concealed.
You must also comply with other laws, including federal law. The federal law you mentioned “The Federal Gun Free School Zone Act,” is covered in 18 U.S. Code § 922 (q)(2) and states that you cannot knowingly possess a gun in a school zone as defined in federal law.
Under exception (B)(ii), of the Act, a Texas LTC holder may legally store their firearm in their personal vehicle in a Texas school zone. The verbiage you cited from the employee handbook, “The Federal Gun Free School Zones Act also requires items to be properly stored and unloaded,” is misleading. The federal law requires that a gun be unloaded if the person does not have an LTC. The language in the handbook is referring to another exception under, (B)(iii), which allows the possession of a firearm in a school zone if the firearm is unloaded and located either in a locked container or locked firearms rack on the vehicle. This exception is irrelevant if you have an LTC.
It is important to note that if you violate the employee handbook, even if it is not a crime, you may be subject to termination.