MacBook – What cable provides support for dual external monitors at 1920×1080 on a MacBook Air

Likely to be the adaptor cable if it's cheap - might not be able to handle the HD throughput (although outputting to HDMI, you'd think it'd be able to).

While this sounds counter-intuitive, try moving around the alignment of the second screen as this might ditch the 'snow'.

Also, have you tried the screen with any other HD inputs?

I am planning to do this with my 2009 Macbook. Not thunderbolt, but a cheap USB or maybe the mini-displayport to 2 splitter--there are a number of options at monoprice in the $40-50 range (probably elsewhere also), might not be fast but it is cheap. There tend to be plenty of reviews for these products, so read them first (many people have done what you want to do).