At the same time, the U.S. Dept of Justice has had a policy for the last 50 years or more (known as the "Petite" Policy, from the name of the first U.S. Supreme Court decision in which it was mentioned) not to exploit the "dual sovereign" doctrine to prosecute someone previously acquitted or convicted in a state prosecution for the same conduct, unless there is a special "federal interest" involved.  The Petite Policy is not judicially-enforceable by the accused; it is solely a matter of discretionary policy by the federal prosecutors.  

Since I already fully answered your question about the "seemingly ... simple" language of the Double Jeopardy Clause, and obviously did not manage to communicate it clearly, let me try to explain another way.  Truth is, your question touches on a very fundamental point about the US Constitution.  Before you can understand the Bill of Rights you have to understand that the Constitution allocates and limits the powers and authority of the federal government vis a vis the states.  When "the People" formed the "United States," they created a system of "federalism" where the states retained a significant measure of sovereignty (that is, the powers ordinarily possessed only by an independent nation).  One of those is the power to make and enforce their own laws (including criminal laws), so long as they don't attempt to override federal law.  The way the Framers would have understood it, a crime under federal law is simply and could not be the "same offence" (to use the original spelling in the Fifth Amendment) as a criminal prohibition on the same conduct under state law.   (An "offense" in other words, is a particular legal violation, not a description of a kind of bad behavior.)

The Constitution does impose certain limits on the power of the states to make criminal laws (look at Article I, section 10), but notice that it does this completely separately from the limits it imposes on Congress in making federal law (Art. I, sec. 9).  So when the Bill of Rights was added, although some of it, like the Fifth Amendment, is written in the passive voice (compare the First: "Congress shall make no law ..."), it is pretty universally agreed that these newly explicit limitations on government power to infringe on human rights were addressed to the power of the federal government only.  Omitting the same limits on the states was not an "exception" to the rules laid down; it was part and parcel of the Framers' political philosophy. (Only in the Fourteenth Amendment, after the Civil War, did the federal Constitution for the first time prohibit the states from depriving persons of life or liberty "without due process of law" (that is, contrary to fundamental American norms of fairness).