Invasion of privacy and false light torts would probably not be applicable here. Very few states have adopted the false light tort because of its conflict with First Amendment principles and there was no agreement or even request to keep the text private.
The copyright issue is trickier. First all, the TOS may provide that the copyright belongs to the text service provider or that there is a license. But, even in the absence of an express license, sending someone a message which is equivalent to sending them a letter, probably gives rise to an implied license that the person to whom it is sent can use the message that arises merely from the act of sending it without restriction or qualification. Implied license and fair use also heavily overlap.
Publishing the text exactly as it was sent to you protects you from defamation liability because it is true.
On the whole it would be extremely unlikely for there to be any legal liability for publishing a text from someone that they sent to you.
Of course, one can imagine exceptions.
If the person receiving the text was in an attorney-client relationship, or priest-parishioner making confession relationship, or was communicating regarding classified national security matters, or there was a non-disclosure agreement in place, among other possibilities, an evidentiary privilege and duty of confidentiality could apply and disclosing the material without the permission of the privilege holder could breach a duty of confidentiality and give rise to liability.
If the picture was a nude picture of a minor, there could be a criminal and/or civil liability issue, and some states have also made posting "revenge porn" a criminal offense and/or a basis for civil liability.
If the disclosure was effectively a way to facilitate insider trading that could be a problem.
If the contents of the text were accurately transmitted but known to be false and were disseminated without disclosure of its falsity for the purpose of defrauding a third party, that could be a problem.
But, no facts that obviously flag any exception are identified in the question. The mere fact that the posting may be embarrassing, or hurt someone's reputation, or was made without someone's express consent, in general, would not be a basis for liability.