Tex. Property Code § 112.009

(a) The signature of the person named as trustee on the writing evidencing the trust or on a separate written acceptance is conclusive evidence that the person accepted the trust. A person named as trustee who exercises power or performs duties under the trust is presumed to have accepted the trust, except that a person named as trustee may engage in the following conduct without accepting the trust:

(1) acting to preserve the trust property if, within a reasonable time after acting, the person gives notice of the rejection of the trust to:

(A) the settlor; or

(B) if the settlor is deceased or incapacitated, all beneficiaries then entitled to receive trust distributions from the trust; and

(2) inspecting or investigating trust property for any purpose, including determining the potential liability of the trust under environmental or other law.

(b) A person named as trustee who does not accept the trust incurs no liability with respect to the trust.

(c) If the person named as the original trustee does not accept the trust or if the person is dead or does not have capacity to act as trustee, the person named as the alternate trustee under the terms of the trust or the person selected as alternate trustee according to a method prescribed in the terms of the trust may accept the trust. If a trustee is not named or if there is no alternate trustee designated or selected in the manner prescribed in the terms of the trust, the court shall appoint a trustee on a petition of any interested person.