I am not able to send text messages, AND text messages from others rejects as "Server Reponse Error" on THEIR phone.

ISSUE:

I receive an error message when sending text messages ("Message not sent"). When SOME people text me they also receive "Server Response Error," on their phone; however, SOME people do not get a rejection and I do receive their text message.  I can receive text messages from SOME AT&T and Verizon friends; I can receive test messages from some iPhone and some Android devices.

• I can place and receive calls. 
• I can send and receive emails.
• I can get AT&T text messages requesting payment.
• AT&T installed a new SIM card last week because I had zero data service. [They said the SIM card went 'bad.') Therefore, it is not a SIM card issue.
• The issue still continues even when my SIM card is placed in another phone (Same model, etc). Therefore, it is not my physical iPhone device.

I've been to 3 different AT&T stores, dealt with AT&T chat, and also dealt with AT&T's 3rd party vendor "Ascend"(?).  They have reset and rebooted my phone. They have updated to the latest soft ware. They have tinkled with the 'network'. Still no luck. Test test test--and still no idea of what the issue is.

AT&T store will not refund me for days of use lost. I use the phone for military reasons and business reasons so data is top priority.

Before I transfer my number to another carrier, does anyone have an idea about the issue?