You're correct that your Texas LTC isn't recognized in California. Handguns are legal to possess in California, and there is no registration requirement until you become a resident. California does have a statute banning the importation of large-capacity magazines (holding more than 10 rounds). Violation is a felony. California also bans threaded barrels and several other features on semi-autos with detachable magazines. Violation is also a felony.
California also bans the carrying of concealed weapons, and loaded weapons. Violation is generally a misdemeanor if the weapon is registered to you by California DOJ. A first time offense with an unregistered weapon is a felony.
California has a statute requiring that weapons kept in vehicles be securely stored. California's CCW statute also applies to vehicles. So long as your weapon is locked in the trunk, you're in compliance with both. Thee is no state statute restricting the storage of ammunition or magazines. But be aware that California's laws regarding "loaded weapons" are kinda goofy and confuse a lot of folks. The general rule (with some notable exceptions) is that a weapon is "loaded" if there is a round in the weapon that can be fired, or placed into a position to be fired, through the normal operation of the weapon. A loaded magazine in the weapon, but with no round chambered, is a loaded weapon. Case law has held that a shotgun buttstock sleeve loaded with rounds did not create a loaded shotgun. But there are exceptions covering certain government buildings, certain portions of Sacramento, gun shows, and cases where a person engages in other felonies, where the weapon is considered loaded if the person possesses the weapon and ammunition. So long as you're not in a place where the exceptions apply, you're fine leaving loaded magazines in the trunk, so long as they're not in the weapon.
California law also has a unique provision that LEO's are authorized to inspect any weapon in a city, or county area where shooting is prohibited (which includes all public roads) to determine if the weapon is loaded.