It wouldn't violate Texas state law to merely communicate with a provider on Eccie. However, if the communication was an offer or acceptance to engage in sexual conduct for a fee, the communication could prove up the offense of prostitution.
The following is the Texas state statute making plain-vanilla prostitution* a crime:
TEXAS PENAL CODE SECTION 43.02. PROSTITUTION.
(a) A person commits an offense if he knowingly:
(1) offers to engage, agrees to engage, or engages in sexual conduct for a fee; or
(2) solicits another in a public place to engage with him in sexual conduct for hire.
(b) An offense is established under Subsection (a)(1) whether the actor is to receive or pay a fee. An offense is established under Subsection (a)(2) whether the actor solicits a person to hire him or offers to hire the person solicited.
(c) An offense under this section is a Class B misdemeanor....
See
Texas Penal Code § 43.02 (emphasis above is mine). A Class B misdemeanor is punishable by up to 180 days in county jail and/or a $2,000 fine.
(*By "plain-vanilla prostitution," I mean prostitution without an aggravating factor, such as it is a second or more commission of the offense, or prostitution involving compulsion or a minor.)
So to summarize, a person can commit the offense of prostitution in any of four ways:
(1) by making an offer to engage in sexual conduct for a fee,
(2) by accepting an offer to engage in sexual conduct for a fee,
(3) by
soliciting a person in a public place to engage with the person in sexual conduct for hire (I take "for hire" to mean the same as "for a fee"), or
(4)
engaging in sexual conduct for a fee.
The
medium to commit the offense is irrelevant. For example, I could commit prostitution by making an offer to a provider to engage in sexual conduct for a fee by transmitting the offer in speech, writing (including an email, text, or note), pictogram, smoke signal, or even telepathically, I guess. Haha.
With the foregoing in mind, if your office computer contains a communication that could be read as either an offer or acceptance to engage in sexual conduct for a fee, it could be used as evidence in making a prostitution case against you.
But in a real-world context, a number of steps would have to be taken for you to be charged. First, a person would have to find the communication. Second, the person would need to transmit the communication to the relevant law enforcement authority. Third, the authority would need to make a report and convey it to the district attorney. Finally, the DA would need to charge you. (In Texas, a person is usually charged with a misdemeanor by a DA signing a charging instrument called an "information.") In my opinion, based on having handled hundreds of prostitution cases in Texas over more than 20 years, the likelihood of all these steps occurring is slim to none. However, my opinion might change if your computer has many offers and acceptances recorded on it.