Wish I could help. I'm having same problem for over 1 week now.
I've been with Online Chat 4 times, called support 3 times and I still outgoing email using Outlook does not work. I've changed all settings as required by COX and tried every variation.
Today I again called Tech Support and their ONLY suggestion was that I should Contact Microsoft Outlook and tell them the issue. According to Cox they are having outgoing email issues with almost ALL of the 3rd party email providers, such as Outlook.
I continue to get the error which states that:
" Personsname"@"server".com on 1/31/2018 11:36 AM
550 5.1.1 relaying not allowed.
Both incoming/outgoing work fine from directly logging into my COX account. However I prefer Outlook since it has integrated, email, calendar, tasks, schedules, etc. all in one piece of software.
Seems like this is COXs issue to fix if they say they support these 3rd party email servers. . Basically the issue is not resolved and I could not get a straight answer if it will be resolved.
Sorry I can not be of help and unfortunately there are many others out there with the same issue. Good luck.