It sounds like you're running into the incompatibility between the SlingPlayer for Windows (Legacy Desktop Version) and your Slingbox 350. Unfortunately, it is true that the Slingbox 350 will not work with this old program. It also appears you are having experiencing poor quality when using Newwatch.Slingbox.com in Google Chrome.
The good news is that even though you are experiencing an issue with viewing via Newwatch.Slingbox.com, you still have two other "SlingPlayer for Web" options.I recommend trying out one of the other options mentioned on this page:
Stay in touch!
Sling Media Forum Moderation Team