I understand the out of sync issue that you are experiencing. It's necessary to confirm if you have this trouble only locally, or, if you also have this issue remotely as well. I recommend trying a different computer just to determine that the issue is not related to the computer or the phone.
We also need to know the upstream connection speed that you have at home and the stream rate that you get when you notice the out of sync trouble. Video/audio quality is limited by the upstream connection available to your HDS-600RS. If there's less than 1 Mbps upstream bandwidth available, you see lower quality. Bandwidth that's 1 Mbps or higher results in higher quality.
Have they tried adjusting from HQ to SQ? Additionally, when first connecting to a Slingbox, it can take 1-2 minutes for the stream to optimize. During this time, you may see that audio/video sync is off. These issues should resolve themselves when the optimization is finished.