https://nyassembly.gov/Rules/?sec=r3#s6 :
"Amendments to a bill shall be made by the member proposing the amendment in the form prescribed by the Clerk of the Assembly and delivered to the Index Clerk.
A copy of each amendment offered by the sponsor of a bill shall be provided by Journal Operations to the chairperson and ranking minority member of any standing committee that previously reported such bill.
Prior to the commencement of debate on a bill to which an amendment is offered by a member other than the introducer, the Clerk of the Assembly shall make copies thereof and cause one copy to be placed on the desk of each member of the House. When there is more than one amendment to a bill and one of such amendments is adopted, any of the other amendments so offered shall, if so requested by the member offering same, be debated and voted upon immediately after such adoption, provided, however, that such other amendments are offered to those provisions of the bill not changed by the adopted amendment and are consistent with the provisions of the adopted amendment.
If a motion to amend prevails, the Clerk of the Assembly shall distribute copies of the bill with the amendment correctly marked thereon together with copies of the detailed description of the amendment to the appropriate Assembly offices for purposes of renumbering and printing the bill as amended.
No amendment shall be allowed to any bill which is not germane to the original object or purpose thereof."
So the process seems on paper simple enough.