OK think this way:
Article VI, Paragraph 2 of the U.S. Constitution is commonly referred to as the Supremacy Clause. It establishes that the federal constitution, and federal law generally, take precedence over state laws, and even state constitutions. It prohibits states from interfering with the federal government's exercise of its constitutional powers, and from assuming any functions that are exclusively entrusted to the federal government. It does not, however, allow the federal government to review or veto state laws before they take effect.
Article IV, Section 1:
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.
Article IV, Section 1 of the United States Constitution, known as the "Full Faith and Credit Clause", addresses the duties that states within the United States have to respect the "public acts, records, and judicial proceedings of every other state." According to the Supreme Court, there is a difference between the credit owed to laws (i.e. legislative measures and common law) as compared to the credit owed to judgments. Judgments are generally entitled to greater respect than laws, in other states. At present, it is widely agreed that this Clause of the Constitution has little impact on a court's choice of law decision, although this Clause of the Constitution was once interpreted differently
Here is why a President may pardon a state crime. A pardon is an executive order addressing a judicial conviction of a person or corporation and that executive order restores both State and Federal constitutional rights and privileges that had been striped of the individual or corporation. For instance John is convicted of a felony in Florida but now resides in Missouri, and operates a business in Kansas. As a result of his Florida conviction he is stripped of certain rights in Florida, Missouri, and and Kansas. i.e. he owns a restaurant and wants to get a liquor license in Kansas. Now in most states John could get a pardon in a sister state (Kansas or Missouri) and pursuant to the full faith and credit clause it would apply everywhere including federal rights. Because his federal rights are impinged by a Florida conviction a federal pardon would be warranted.
The DOJ opinion that a President can't issue a pardon for state crimes serves the DOJ well as it doesn't have to process tens of thousands of pardon applications and it is based upon a bad interpretation of the Hatch Act.
The Hatch Act of 1939, officially An Act to Prevent Pernicious Political Activities, is a United States federal law whose main provision prohibits employees in the executive branch of the federal government, except the president, vice-president, and certain designated high-level officials, from engaging in some forms of political activity. It went into law on August 2, 1939. The law was named for Senator Carl Hatch of New Mexico. It was most recently amended in 2012
What the DOJ asserted is that restoration of the right to vote is a political act, and while the President might issue a pardon sua ponte but government employees were prohibited from work on the pardon including investigating, reading and filing recommendations, searching official records and typing the document.
So if a President really wants to pardon someone for a crime in NY State he can type it on his PC, print it and sign it