The following is my interpretation, for what that is worth, of Texas Penal Code.
According to The Texas Statutes, Penal Code, a person who has a CHL and is carrying an authorized handgun can also carry a club or a knife (or a sword) that would otherwise be classified as an illegal knife.
Here are the applicable excerpts from the Penal Code:
§ 46.02. Unlawful Carrying Weapons
(a) A person commits an offense if he intentionally,
knowingly, or recklessly carries on or about his person a
handgun, illegal knife, or club...
§ 46.01. DEFINITIONS. In this chapter:
(1) "Club" means an instrument that is specially
designed, made, or adapted for the purpose of inflicting
serious bodily injury or death by striking a person with the
instrument, and includes but is not limited to the
(5) "Handgun" means any firearm that is designed, made,
or adapted to be fired with one hand.
(6) "Illegal knife" means a:
(A) knife with a blade over five and one-half inches;
(B) hand instrument designed to cut or stab another by being thrown;
(C) dagger, including but not limited to a dirk, stilletto, and poniard;
(D) bowie knife;
(E) sword; or
§ 46.15. Nonapplicability
(b) Section 46.02 does not apply to a person who:
(6) is carrying a concealed handgun and a valid
license issued under Subchapter H, Chapter 411,
Government Code, to carry a concealed handgun of the
same category as the handgun the person is carrying;
When you think about it, this makes perfect sense. If you are legally allowed to carry a firearm, and you are in a situation where it is legal to carry, who cares if you are also carrying something else that is classified as a weapon?
On the other hand, if you are not legally allowed to carry a firearm, or if you are in a situation where it is not legal to carry, you probably should not have other weapons on your person.