Landlord Requesting New Lease After Move-in, Claiming Current Lease Isn't "executed"

Hi Reddit legal advice,

My partner and I recently moved into a luxury apartment building in NJ. We signed a 90+ page lease agreement and submitted certified checks for our security deposit, first month's rent, and a amenity fees. We moved in during early June for a one-year lease term.

Management is now requesting us to re-sign an entirely new 90+ page lease. They claimed our original signed lease was "missing a required amendment related to buyout and cancellation fees." I compared both documents and found that our signed lease does contain buyout/cancellation terms. While one section had a blank for the numerical amount, another section in the same addendum clearly states the buyout fee is "2x the monthly rent and 60 day notice." This formula yields the same amount explicitly stated in the new lease they presented. So, the terms are present and consistent.

After speaking with management, they are now claiming our current lease is "not an executed lease" and that they are "holding the certified checks have not applied our payment" despite them having provided us with the keys and allowing us to move in.

This feels like an attempt to invalidate our existing, acted-upon agreement. My questions for the subreddit:

1. Is our current lease legally binding in NJ, given we signed it, paid, and moved in?

2. What are the legal implications of them saying our current lease isn't "executed"?

3. What recourse do we have if they won't acknowledge our current lease and could we seek concessions for this unusual request and their misrepresentation

4. Should we sign their new lease, especially under this pressure?

Any advice, especially from those familiar with NJ landlord-tenant law, is greatly appreciated.

Location: New Jersey