My Outgoing Email setting are working
Here was my experience with Cox. I've been with Cox for almost 20 years. Sometimes my out going emails are sent sometime they are not sent. I was frustrated the same way you are. I need to access my email via my cell phone. I always had the outgoing server settings right. They told me to talk to Apple as well. Apple said its not them. My frustrations gotten bigger and was thinking about switching.
Then I changed my password one day and saw my username was different than what I entered on my phone. See, Cox always populated @cox.net to my username. So I thought I might as well start entering that in and that was my mistake. Please just enter your username then your password and you should be able to send emails. For example, username@cox.net (wrong). username (right). Enter your username without the domain then enter your password and it should work.
Please try that.