Interesting. My neighbor and I also were prompted by Cox to change email password because of "Suspicious activity". A few days later, Cox blocked me from sending any email because of "Spam like behavior". A completely outrageous and false accusation. Sorry, I'm digressing. It is starting to look like Cox was hacked but they are not fessing up to it.
Yes, the same password you created needs to be entered for the sending or SMTP server and it needs to be done on every device you use for email.