Thanks for using our forums. We are sorry that you are having troubles with your Slingbox when using HDMI cables. It sounds like you are running in to HDCP or SOC coming from your cable box. These two things are forms of copyright protection, coming from the cable box. In essence, what is happening is that the cable box is recognizing that something other than the cable box is connected and it thinks it is trying to steal the cable so it is shutting down the signal. The fix for this would be to remove all HDMI cables completely and use component cables as a passthrough. Below you will find more information about HDCP. The article does state for the 500, but it still applies to the 350.
Hope this helps,
The Sling Moderation Team