bw100 - 5:41 pm on Jun 11, 2011 (gmt 0)

If you're still a resident of CT for tax purposes, but your Amazon 1099 comes to an address in NH (which has no state income tax), then you would still legally owe CT income tax on that earning.

This is absolutely correct. As well as Federal tax, which return would also show your Connecticut address. And the Feds and the States share data.

But when you pay it, and CT sees a bunch of Amazon 1099s coming down the pipeline, aren't they just going to claim Amazon's operating in their state

Yes, and they would be correct ...
You can run, but you can't hide.