Ideally, you should do background checks in both Michigan AND New York, to see if the charge comes up. Yes, you will have to pay fees to get these records. In Michigan, you will have to pay to get fingerprinted, then you will have to provide these fingerprints to the Michigan State Police, along with a fee for them to process a criminal background check on you. You should also contact a New York-area criminal defense attorney to confirm how the process is done in New York.
Although you can do this yourself, you can also hire a reputable private investigator to look into your record. They are often not as expensive as you might imagine.