I've seen the audio/video out of sync problem with Dazzle many times. Don't know how to solve the problem. But Dazzle has tech support, so it is a good idea to contact them.

I suggest you to get an HD capture card. See more details in the following thread:
Record Xbox 360 gameplay with Pinnacle Studio MovieBox Ultimate 14 HD?

AverTV HD DVR with an HDMI splitter will cost slightly more than $100. Note that you will need a very powerful computer to capture HD videos with it.