How do I block my phone number when sending texts?

Do you have iMessage set up? I was assuming you did. If so Settings -> Messages -> Receive At -> Caller ID

Note that if sending a regular SMS text through your carrier, I do not know of any way to block your number in that case - the only ID a carrier has for you in their SMS system is your phone number.

iMessage, of course, does not go through your carriers text servers, but goes through Apple's messaging servers instead, so it can allow that option of what the recipient sees as the "from" ID.

Put something else in your caller ID setting (an email address) under messaging and make it your default caller id for messages.