Changing Public IP Address

yes I do

Other than getting tossed offline while streaming from a particular site, all other Internet browsing and services work?

What do you suspect is happening with your IP address since it had leaked on a stream?  What was the stream to have leaked your address?

I stream on twitch, its my job, and my ip got leaked when one of my viewers got in the same game as me and leaked my ip in my discord server. When they ddos me my internet completely shuts down. 

So you're not getting tossed offline while streaming but instead your modem is being overwhelmed with bogus requests.

I'm surprised Cox couldn't detect this because a DoS not only affects an address but also the entire network.

What you could do is send an email to cox.help@cox.com with the following:
 - Full Name
 - Complete Address
 - URL of This Post

ok will do