I understand. We can also do a phone call if you'd like (obviously this service/company charges a fee for that). Mr. George, I noticed you haven't answered my other questions.
Have you updated the DNS records (MX, and if necessary A records) to reflect the new IP address(es) provided by Time Warner for your domain?
Your previous ISP gave you 5 IP addresses
Let's say you had: 18.104.22.168 assigned to your Exchange Server. That means you have a DNS MX record pointing to 22.214.171.124
If your new ISP Time Warner gave you 5 new IP addresses...
Let's say you now assigned the IP address 126.96.36.199 to your Exchange server, then you must update that DNS record (MX) to point to the new IP address.
Then of course you could also be doing NAT (Network Address Translation) which means your firewall has an IP address, and all devices behind that firewall share that same IP address including your Exchange Server. This is because NAT translates that public IP to an internal IP based on port/service. Ex: Any https requests coming through to the public IP may be directed to your Exchange Server's Outlook Web Access site.
To really help you I need you to answer my questions and fill me in on your network topology. Are you doing NAT? Are you assigning one of the 5 static IPs to your Exchange Server? Did you update your DNS?