Yes, do an affidavit of service, but keep it in your file--you don't have to file it in the County Clerk's office. Service of a copy of the judgment with notice of entry starts his appeal time, so you want to be able to prove when you served him.
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You don't have to file an Affidavit of Service.
I am a former federal and State prosecutor and have been handling criminal defense and personal injury cases for over 17 years. The above answer, and any follow up comments or emails, is for informational purposes only and not meant as legal advice.