You posted your "contract" in conversations. I don't do private discussions. You can post it here. Bottom line, though, it's not a "contract." It's your agreement to abide by the law.
Yes, lifetime registration "without judicial review" was declared unconstitutional by the SC Supreme Court earlier this year. However, since you no longer live in SC, that seems to be irrelevant since it's not binding on NY.
You live in NY and are subject to NY's Sex Offender Registration Act:
Sex Offender Registration Act - NYS DCJS
Note that the statute has provision for petitioning the court for relief or modification of your registration requirement (Section 168-o).
See also:
Sex Offender Petition for Relief or Modification | NY CourtHelp (nyplace.org)
I haven't been able to find any DIY forms online but I did find this article regarding the process. This is not a referral. Any criminal defense attorney can help you.
How to Modify or Reduce Sex Offender (SORA) Level in NYS | King Law (robertkinglawfirm.com)
You might stop in at your local court and see if there is some kind of forms and instructions packet. If not, you might find samples at a law library (google NY law library for one near you).
Unfortunately, if you cannot figure out how to do this yourself you'll have to wait until you can afford to hire an attorney.