Although not explicitly called out in the Constitution, the source of our dual sovereignty federation is the 10 Amendment:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
The purpose of the Tenth Amendment was to protect the rights of the States and, more importantly, the citizens from an overbearing and intrusive Federal government. In it's roughly 230 years of existence, this single sentence has been at the center of probably hundreds if not thousands of court cases reaching all the way to the Supreme Court. Specifically, dual sovereignty has been tested in the Supreme Court numerous times and still stands.
But dual sovereignty can be thought of in a different manner also. I can be looked at as additional protection for states with smaller populations along with the electoral college and the organization of Congress with one house representing the people and the second giving each state equal voting power. These protections would come into play if the Federal government attempted to pass laws contrary to our Constitution, the Green New Deal and socialism for instance.
With the initiation of a bill to implement bad legislation such as this, the Senate (with each state having equal voting power) can block bad legislative bills from becoming law. Since there are more states tending to be more conservative than there are liberal states, it would appear bad legislation would never pass in the current Senate.
If it did, under our dual sovereignty federation, these more conservative small population states could simply decide not to obey the new Federal law. It's easy to see as many as twenty to twenty-five states taking this stance. With something on the order of half of the states in the Federation not complying with the new law it will be rendered useless.
For those patriots in states that would accept a new socialist government structure, dual sovereignty protects the individual citizen by allowing them simply to move and retain their Federal citizenship. Within a short period of time, the citizen that has moved will establish residency in their new state and become a citizen of that state. Similarly, those seeking the "protection" of socialism would be able to relocate to those newly established socialist states. I imagine the migration within the Federation would reach historic levels.
From the dual sovereignty system the doctrine of anti-commandeering has been established. Under this doctrine, the Federal government cannot not force states to comply with Federal law. Although socialism will never work long term, you have to remember Margaret Thatcher's quote:
The problem with socialism is that you eventually run out of other people's money.
The more people under the socialist system, the more money there will be to keep it going allowing it to last longer than it would with a smaller number of people. As a result, the liberal Federal government will attempt to override the doctrine of anti-commandeering and force the states to join the new socialist Federation.
When the Federal government attempts to force issues such as this, Alexander Hamilton in Federalist 46, wrote:
A correspondence would be opened. Plans of resistance would be concerted. One spirit would animate and conduct the whole. The same combinations, in short, would result from an apprehension of the federal, as was produced by the dread of a foreign, yoke; and unless the projected innovations should be voluntarily renounced, the same appeal to a trial of force would be made in the one case as was made in the other.
Here, Hamilton is indicating the states should react to an overreach of the Federal government in the same manner as the Federal government would react to an overreach of a foreign power. Hamilton is suggesting the two sovereign governments (Federal and state) first try to diplomatically to resolve the disagreement and as a last resort move to a "trial of force" or armed engagement. Let's hope it never comes to that.
The founding fathers, at least Hamilton, saw dual sovereignty as a set of strong states linked together by a less powerful Federal government. Liberals are already attacking our electoral college because it goes against our democracy, either unaware or ignoring the fact the United States is not a democracy. These attacks are not based on the good of the country as a whole but directed toward the attainment of political power. Sooner or later, dual sovereignty and eventually the existence of the Senate will suffer these same attacks because they represent the same limitation on their attainment of power. These institutions represent a limitation of politicians at the Federal level taking power and subsequently freedoms from the States and the citizens. The last 150 to 200 years has seen a huge expansion in the power of the Federal government taking authority from the sovereign states, let's hope the next 100 to 150 years can be spent putting these powers back in the hands of the States (and the people) where they were intended.