One of the major flaws of the Articles of Confederation was that it didn't establish any mechanism of governance to deal with interstate disputes or anyway to enforce laws or agreements across the colonies. Because the Articles of Confederation established such a weak central authority the Constitution was explicitly written to correct this problem, and establish a government that could govern the states in an effective way so that they could function as a nation.
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That's where they were able to find the resources they needed for economic growth like minerals such as iron and gold. They didn't settle somewhere else because no other land had the resources they were looking for.
The Byzantine Greeks called themselves Rhomaoi (Romans) and their Empire the Roman Empire. They saw themselves as the continuation of the Roman Empire and never used the term "Byzantine" to describe themselves. There was no one clear period when the Eastern Roman Empire became the Byzantine Empire.
