We just switched over from Time Warner 3 days ago and since install can not access any of our mail. We have AOL, Gmail, Apple Icloud (old me accounts) and never had any issues but the second we hooked up to Verizon unable to send or receive... 3 days of this is crazy. I've called and spoke with people about this first time it was an outage next time it was hum must be having issues with all your email servers... REALLY all at once the same issue the minute switched...
What email client are you using? What error(s) are you getting? Is the internet otherwise working correctly on this computer?
Just going out on a whim here and saying either a firewall on your computer or within your modem is blocking some part of the connection process, or you need to change the mail client to use secure methods and ports. Verizon is a bit more restrictive with outgoing e-mail port 25 (SMTP) in that traffic outbound on that port is filtered to reduce spam and open proxies from making a mess.
But yeah, we need error codes from the client you use.