First, any licensee (real estate agent) must disclose his licensure in brokerage to potential buyers, and it is best to get written proof that it was disclosed prior to execution of a contract. See 1101.652(b)(16) in particular.
Second, use of a multiple listing service (MLS) is reserved to brokers, not agents. So, in order to list the property in MLS, the "agent" must sell and all listings belong to the broker. So, if MLS is involved, then, no, the agent cannot sell without his broker.
Third, the real estate licensee must follow the rules established by his sponsoring broker for conduct of business. If the broker policy is that all sales must be done at arm's length and no agent may list his own property, then you have your answer. Ask the broker what the rules are.
Lastly, if the real estate licensee has passed through all of the above tests: disclosing, not using MLS, and no broker policy against self sales, then your question becomes, "Can a licensee sell privately to someone?" I can't find anything in TRELA that disallows it.
TRELA says if you sell or offer to sell on behalf of someone else (for compensation), then you must be licensed. To be licensed you must be sponsored by a broker. And you must follow all the brokers rules (including MLS rules). If you are offering to sell your own property (and the broker has no rules against it), how can you be restricted?
Here is 1101.004. "A person acts as a broker or salesperson under this chapter if the person, with the expectation of receiving valuable consideration, directly or indirectly performs or offers, attempts, or agrees to perform **for another person** any act described by Section 1101.002(1), as a part of a transaction or as an entire transaction."