Is it considered harassment to send anonymous texts to an ex-lover’s wife telling her that he is cheating?

This is a touchy area. Here is the Texas statute:

§ 42.07. HARASSMENT. (a) A person commits an offense if, with intent to harass, annoy, alarm, abuse, torment, or embarrass another, he: (1) initiates communication by telephone, in writing, or by electronic communication and in the course of the communication makes a comment, request, suggestion, or proposal that is obscene; (2) threatens, by telephone, in writing, or by electronic communication, in a manner reasonably likely to alarm the person receiving the threat, to inflict bodily injury on the person or to commit a felony against the person, a member of his family or household, or his property; (3) conveys, in a manner reasonably likely to alarm the person receiving the report, a false report, which is known by the conveyor to be false, that another person has suffered death or serious bodily injury; (4) causes the telephone of another to ring repeatedly or makes repeated telephone communications anonymously or in a manner reasonably likely to harass, annoy, alarm, abuse, torment, embarrass, or offend another; (5) makes a telephone call and intentionally fails to hang up or disengage the connection; (6) knowingly permits a telephone under the person's control to be used by another to commit an offense under this section; or (7) sends repeated electronic communications in a manner reasonably likely to harass, annoy, alarm, abuse, torment, embarrass, or offend another.

Electronic data (aka texting) is considered to be part of this statute. While I can see that you think you have a defense because you believe it to be true, I would stop immediately. It is a fine line here.