No, you do not need to contact the Tax Collector personally to notify of an address change. At the closing, when the deed is changed, the Tax Collector's Office is notified.
It takes six to eight weeks after the closing for a new owner's name to be officially recorded on the property deed and in the municipal tax system, so the property may still be listed under the previous owner's name. If it is less than 8 weeks since you became the owner, you must give it more time. If after 8 weeks it is still not showing you as the property owner, contact the Engineering Office so that the deed can be updated.
However, if you just want to change the mailing address for property tax/sewer bills, you must notify the Tax Assessor's Office at (609) 890-3654.