I'm sorry about the loss of your mother.
You're not legally required to use an FFL to ship the guns to yourself. The problem would for your brother to persuade UPS or FedEx to accept them for shipment to you. I've heard several stories of them refusing to ship to an unlicensed person.
As I read the law, your brother can ship them by U.S. mail to a Texas FFL, and you can claim them without going through NICS (with a CHL). FFLs charge anywhere from $10 to $25 for that service.
Were these guns ever registered in New York? I don't know how things work there. In most states that have registration, the registered owner may be required to provide proof of the disposition.
http://www.atf.gov/firearms/faq/faq2.htm#b7
(B7) May a nonlicensee ship a firearm through the U.S. Postal Service?
A nonlicensee may not transfer a firearm to a non-licensed resident of another State. A nonlicensee may mail a shotgun or rifle to a resident of his or her own State or to a licensee in any State. The Postal Service recommends that long guns be sent by registered mail and that no marking of any kind which would indicate the nature of the contents be placed on the outside of any parcel containing firearms.
- Jim