Doesn't matter if you move - you still have to pay the taxes no matter where you live.

You missed the entire point of the link. Let me see if I can explain it to you. When Maryland implemented its millionaire tax, hoping to increase revenue, it actually lost money because the millionaires moved to other states that did not have a millionaire's tax. The millionaires did not have to move out of the country.
Do you get it now?

I would like to try living outside the USA for a year or two sometime - but the prospect of having to pay taxes twice has kept me from really trying to pull it off.

Has the USA passed new tax laws since 2001? I lived and worked in England for 2 years from 1999 to 2001. At that time, the UK had a tax treaty with the US. This meant that if I was paying taxes on income earned in the UK while living in the UK, I did not have to pay taxes again to the US on that income. Doing my US taxes while I was resident in the UK was a real PITA - I had to fill out all of these extra forms proving that I was resident in the UK for tax purposes by passing what they called the physical presence test (which meant that I was living in the UK for at least 330 days that year).

A brief GOOGLE search seems to indicate that this is still the case:

https://www.irs.gov/Individuals/International-Taxpayers/Foreign-Earned-Income-Exclusion---Physical-Presence-Test