I don't want to fill this form out wrong and end up paying fines for it.
What form? You plead not guilty and fix the problem then submit proof as outlined below. You need to call the court to find out if the ticket and proof can be mailed in (which some courts allow) or if you must appear when required to.
Did you get a Statement of Correction with the ticket? If so then there is nothing for YOU to fill out. You need to have a police officer from any department inspect your vehicle to verify that it has been fixed. He will sign off on the form. That needs to be done by sunset of the next business day after the ticket was issued. There are alternate means of providing said proof as specified in the below excerpt from the Vehicle and Traffic Law:
5. (a) Acceptable proof of repair or adjustment shall consist of
submission to the court on or before the return date of the summons
either:
(i) a statement of correction from an officially designated state
inspection station duly executed by the person performing or making such
inspection and bearing the facility number of the state inspection
station, or
(ii) a statement of correction from an automobile repair shop on the
letterhead of such repair shop duly executed by the person who made the
correction, or
(iii) a statement of correction from any registrant having more than
twenty-five vehicles registered and having a fleet maintenance program
administered by the registrant, duly executed by the person performing
or making such correction and countersigned by the fleet maintenance
supervisor, or
(iv) a signed statement of any police officer that the necessary
corrections have been made, or
(v) evidence acceptable to the court from any person that he or she
completed the repair together with proof of purchase of the equipment
needed for the repair, or
