This issue is likely related to your outdated modem/router combo. A new wireless router/Access Point will likely solve your issue.
Unfortunately, as far as CenturyLink is concerned that is a current, up to date modem. I have the same one, and have constant issues with it overheating. I have a fan blowing directly on it year round, because even in winter it gets too hot. I have Wi-Fi disabled on it, and use different routers as WAPs on my network. I've complained to CL about the poor hardware, but they have no other options available. Those Actiontec routers are cheap, which is probably why CL sticks with them.
That said, as long as it hasn't locked up I never have any streaming issues with any of my Roku players or other Internet devices. I don't have the Stick Plus, but everything I have works fine.