But moving past that for a moment, why doesn't the state of New York do it's own investigation of Adams? It seems possible that if Adams has broken some federal corruption laws, he's broken state corruption laws as well. Or maybe not.

The state and federal public corruption laws are very different. Links at the bottom.

I am not a criminal lawyer. I think the key difference is that the NY state statutes seem to require an "agreement" as a required element. There's numerous versions of the same crime depending on the amount involved, but here's what it says:

A public servant is guilty of bribe receiving in the second degree when he solicits, accepts or agrees to accept any benefit valued in excess of ten thousand dollars from another person upon an agreement or understanding that his vote, opinion, judgment, action, decision or exercise of discretion as a public servant will thereby be influenced.

Looking at that, the elements that need to be proven are: i) public servant; ii) solicitation, acceptance or agreement to accept; iii) any benefit > $10K; iv) agreement or understanding to have an action influenced.

The federal statute kicks in when a government official:

corruptly solicits or demands for the benefit of any person, or accepts or agrees to accept, anything of value from any person, intending to be influenced or rewarded in connection with any business, transaction, or series of transactions of such organization, government, or agency involving any thing of value of $5,000 or more

Those elements are: i) government official; ii) accepts or agrees to accept; iii) thing of value; iv) intent to be influenced in an action.

The key difference is in the fourth element. The state law requires the prosecution to prove the existence of an agreement or understanding, while the federal law only requires proof that the official intended to be influenced. It's probably much, much harder to prove the former than the latter, because the existence of an agreement or understanding will require proving a "meeting of the minds" between the briber and bribee - while the federal statute only requires proving the mental state of the person receiving the bribe.

law.cornell.edu: 18 U.S. Code § 666 - Theft or bribery concerning programs receiving Federal funds | U.S. Code | US Law
ypdcrime.com: Article 496 | NYS Penal Law | Corrupting Government
ypdcrime.com: Article 200 Penal Law