To the best of my knowledge:
The firearm is not changing hands, and therefore a transfer does not take place.
It therefore is legal to ship it to yourself (or someone ship it to you) but ONLY by ground freight service.
FedEx and UPS can handle this for you (or her). DO NOT DECLARE IT IS A FIREARM, FedEx will require (as part of their regulations for operation) that it ship to an FFL, which is NOT needed in this case. Do, however, get insurance on it. As you already know, do not ship ammo with it.
It must be in a hard-sided case that is lockable, inside another box (shipper's box).
Again, since no transfer takes place, no FFL is needed.
Again, it MUST ship ground only.
DO NOT send it thru the Postal Service, use a freight forwarding service (such as FedEx or UPS).
THE ONLY RESTRICTION would be any such restrictions on the firearm in the destination state. For instance, a Barret .50BMG rifle is illegal in California, so one cannot ship it to California. I am not aware of any restrictions in Texas over handguns, but I am not an expert on Texas law, you will have to research this yourself, and if you ask: Is this gun legal in Texas and the answer is YES, then there are no restrictions.
This question gets asked almost every gun class that I teach, and the answer is always the same: if there is no change of ownership (especially if the origin or destination require firearms registration) then no FFL is required because no transfer takes place; it MUST ship ground freight only; it MUST be legal to possess in the destination state and city; it must ship without ammunition; restrictions imposed by the Interstate Commerce in Firearms Act do not apply since the firearm is not an article of commerce (no transfer); the Gun Control Act of 1968 likewise does not apply for the same reason (no change of ownership).
Only local laws and restrictions (if any) in the destination jurisdiction would apply.
Hope that helps.