Can't send to certain providers

I can't send to q.com or centurylink through Cox.  However, if I turn off my service provider COX, the mail goes right through; so the problem is not at my end, it is COX.  They claim it is not.  For a short period of time I could send to these sites, but now I can't again.  I do use Apple, but it sends fine as long as I don't use COX.  Why do I stick with this provider.  I don't like change, but I do appreciate good service which I get from Apple and don't get when I try to call or chat with COX.  I was told at one point that my account was flagged as spam.  That might have been because I was contacting large numbers of people for a school reunion, but supposedly, that was taken care of.  My lawyer also can't get my email which is my biggest frustration.  I can send if I don't use COX.  Just turn off my home Wifi and it works.