Could the national government solve problems between the states?

According to the Constitution the answer is yes. The federal government is the guardian of the Constitution and Article IV the Constitution establishes a series of guidelines and principles that dictate the duties, rights and powers of states towards the federal government and to each other. Two sections are of special importance:

Full Faith and Credit shall be given in each State to the public Acts, Records, and judicial Proceedings of every other State. And the Congress may by general Laws prescribe the Manner in which such Acts, Records and Proceedings shall be proved, and the Effect thereof.

This section is known as the "Full Faith and Credit Clause" and it means for example that a couple who married in Colorado and then moves to Texas will still be legally married in Texas which is bound by this clause to accept the marriage certificate from Colorado as a valid legal document.

The Citizens of each State shall be entitled to all Privileges and Immunities of Citizens in the several States.

This Section is known as the Privileges and Immunities Clause and it means for example that American citizens to move from Texas to California have to be granted all the rights and privileges that the state of California has granted to its native-born residents.