@Nick Brockelman, I can attest to this personally - I'm excited to have a house under contract, but had to go quite a bit higher than asking price due to other bidders and am waiving the appraisal contingency. 

One of the first things to explore would be selling that property contingent on you finding a suitable replacement property for your 1031.  

Yoh can put offer on the house contingent that 1 unit is vacant upon closing date.Or like mentioned, ask when lease end is for tenants.

Has anyone ever made a “cash” offer contingent on financing?

Homes are regularly going for 10-20% over asking with no contingencies.

We will see if I over-bid when we get the appraisal - I'm protected by a couple contingencies to walk away from the deal if need be.

Maybe if you have some sort of contingency clause you can get out of the deal without losing the deposit then go that route.

There was an appraisal contingency and inspecton contingency and quite frankly we were shocked when we got the call from the listing agent.

Market value and sales trend in NY are also something to consider are the units being sold based on reasonable comps arms length transaction or are there an excessive amount of cash sales with no appraisal/contingencies.

Amend the contract to extend your contingency period(s) and don't spend anymore money until AFTER the property is properly parceled.