The following is my interpretation, for what that is worth, of Texas Penal Code.
Summary
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
following:
(A) blackjack;
(B) nightstick;
(C) mace;
(D) tomahawk.
(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
(F) spear.
§ 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;
Comment
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.