Problem with sending email

Since 4/24 we have not been able to send emails using our Cox accounts through the mail app while using data on our iPhones. We can send through gmail & other accounts or while using wifi. We have the most current IOS & have confirmed the email settings are correct. I have contacted Apple, T-mobile, and Cox multiple times. Everyone points at the other. However, Cox updated their outgoing email server last night when this started. It appears to be a Cox problem, but they will only try to resolve if I pay for Cox Complete Care to reach a real tech, instead of a gatekeeper. Been with Cox for 2+ decades & I do not appreciate this "service". So how are you going to fix it?