No carrier will allow anonymous SMS texting. In the USA, carriers simply refuse to transmit messages where someone has used a hack or other means to strip the sending number from the message.
You could pay a service to get an SMS short code for sending, if you wished to keep your phone number private (but, short code service providers are not cheap - why mostly only corporations use them legitimately).
To avoid using your phone number, you can send an SMS via one of your email accounts using the recipients SMS email gateway (no phone number shown, but they will see your sending email address). Or you can use a service like WhatsApp and an alias, although the recipient would also need to be using the same service. And Apple's iMessage allows use of an email in the sent from field, instead of a phone number, but again, the recipient must be an iMessage user.