The Texas Occupation code is irrelevant to you since you are not in the occupation of being a plumber. Instead, as a homeowner, you are subject to your local building codes. This is the Houston Plumbing Code. You particularly need to pay attention to the obligations and exemptions of homeowners. §102.10 says
In accordance with the Plumbing License Law, this code shall not
prevent any homeowner from installing and maintaining plumbing in a
building owned and occupied by him as his homestead if done in
compliance with the requirements of all applicable state-adopted codes
and ordinances of this jurisdiction. Such privilege does not grant the
right to violate any of the provisions of this code or state-adopted
codes, nor shall it be construed as exempting any such property owner
from obtaining a permit and paying the required fees therefor, except
for work that is exempt from permitting under this code.
In other words, if a permit is required, you must get the permit and pay the fee, except not all work requires a permit.
§104.2 specifies what constitutes exempt work:
A permit shall not be required for the following: (1) The stopping of
leaks in drains, or soil, waste, or vent pipe, provided, however, that
if a trap, drain pipe, or soil, waste, or vent pipe becomes defective
and it becomes necessary to remove and replace the same it with new
material, the same it shall be considered as new work and for which a
permit shall be procured and inspection made as provided in this code.
(2) The clearing of stoppages, including the removal and
reinstallation of bathroom or kitchen faucets or water closets, or the
repairing of leaks in pipes, valves, or fixtures, provided such
repairs do not involve or require the replacement or rearrangement of
valves, pipes, or fixtures. Exemption from the permit requirements of
this code shall not be deemed to grant authorization for work to be
done in violation of the provisions of the this code or other laws or
ordinances of this jurisdiction. This section shall be construed in a
manner that is consistent with the Plumbing License Law, and no
provision herein shall be construed to exempt work for which a permit
is required to be obtained from this jurisdiction.
The main reason to care about this document is that municipalities are entitled to and often do modify the "Uniform" codes, and it is pointless to argue "The Code says..." when their reply is "We require that...".