I think you will be fine, an ACD is not a conviction.
This answer does not constitute legal advice and you should contact an attorney to confirm or research further any statements made in this answer. Any statements of fact or law I have made in this answer pertain solely to New York State and should not be relied upon in any way in any other jurisdiction.
The cases would not have shown up if they were sealed.
The above answer, and any follow up comments or emails is for informational purposes only and not meant as legal advice.
You have certain rights under the Fair Credit Reporting Act if these sealed records pop up on your background report. For example, if the background company reports old sealed records, you may have a claim in Federal Court for lost wages (if you lose the job opportunity) because the reporting company should have maintained reasonable procedures to guarantee maximum possible accuracy. You could also have claim against the employer if the employer fails to provide you with a copy of the report plus a summary of consumer rights (which would tell you how to dispute the report) BEFORE it makes a decision not to hire you. Hopefully these old records do not pop up.