It's not clear whether you mean that the entire agreement is carried out by text message. If you have a paper or electronic document stating what the parties will do, that is the agreement, and signatures are a conventional form of proof that there is an agreement. A handshake or a verbal statement – or text mesage – could also serve as evidence of the agreement, though there could be problems with the quality of the evidence (e.g. how do the witnesses know which piece of paper you agreed to).
There is not a huge body of law surrounding text messages (and apparently none regarding text messages and contracts). We know that a text message is not a "printed receipt" (Shlahtichman v. 800 Contacts), and that a text message is a "call" w.r.t. the Do-Not-Call law (Campbell-Ewald Company v. Jose Gomez, Keating v. Nelnet). The closest that I have been able to come in terms of a decision about whether a text message is "written" is Commonwealth v. Mulgrave 472 Mass. 170, which states that
While Massachusetts appellate courts have yet to approve admission of
text messages or any other writing under the spontaneous utterance exception to the hearsay rule...
The wording "any other writing" implies that the court believes text messages to be "writing", which of course it is if you look at the plain meaning of the word "write".
Contracts can be formed and signed by email: 15 USC 7001 states that
a signature, contract, or other record relating to such transaction
may not be denied legal effect, validity, or enforceability solely
because it is in electronic form....a contract relating to such
transaction may not be denied legal effect, validity, or
enforceability solely because an electronic signature or electronic
record was used in its formation
so the fact of electronic writing does not make the contract non-written.
There must be a venerable and well-known rationale behind the written / oral asymmetry, which presumably has to do with the volatility of memory which would be front and center in a dispute about oral contracts.
FRE 1001 "clarifies" that a "writing" consists of letters, words, numbers, or their equivalent set down in any form, and that a "recording" consists of letters, words, numbers, or their equivalent recorded in any manner. 15 USC 7001 also states that
An oral communication or a recording of an oral communication shall
not qualify as an electronic record for purposes of this subsection
except as otherwise provided under applicable law
So a recording of an oral contract would be useful to prove that there was an agreement, but would not change the fact that the contract is oral.