You are bound by the treaties your country signed.
Yes: You, and the states, and their courts, are bound by them (to the extent they are clear or were implemented by federal enabling legislation).
In fact, they have more legal weight in the US than laws passed by your own Congress.
NO! They have EXACTLY the same weight as federal law. Both treaties and federal law are trumped by the Constitution, and both are also creatures of Congress, They can be modulated, and destroyed (at least in how they are effective within the country) by congressional action.
The idea that they're any stronger or more permanent than federal legislation comes from a (very common) misreading of the Supremacy Clause:
This Constitution, and the Laws of the United States which shall be made in pursuance thereof; and all treaties made, or which shall be made, under the authority of the United States, shall be the supreme law of the land; and the judges in every state shall be bound thereby, anything in the constitution or laws of any state to the contrary notwithstanding.
This says that the Constitution, Federal Law, and Treaties trump state law in state and federal courts. It says nothing about the relative power among the three.
The misreading is to interpret "all treaties made ... shall be the supreme law of the land ..." to mean that treaties effectively amend the constitution. This is wrong. You can see it by noticing the same kind of misreading also makes federal law equivalent to a constitutional amendment - which it clearly is not.
In fact the Supreme Court has spoken on the relation between the Constitution and treaties: In Reid v. Covert, 354 U.S. 1 (1957), the Supreme Court held stated that the U.S. Constitution supersedes international treaties ratified by the U.S. Senate.
Treaties are abrogated, at the federal level, all the time, and there are a number of mechanisms for doing so.