In New York, we reccomend that you hire an attorney to handle the sale and purchase of a home. As a real estate agent it is impossible to answer this question with more that speculation. You state that you "entered into contrat on 4/14". Was that handled by an attorney? The best advice I can give is to consult your attorney for legal advice. Depending on the wording in the contract an attorney can advise you in which direction to go.
I wish you rhe best of luck.
Barbara Carter Exit Realty Services New Paltz NY
A lawyer will advise you on how to tie up the property. However, a lawyer may tell you (as lawyers have told me more than once): You may prevail in court. But it will be very expensive. My comments aren't legal advice, just an observation on what you may encounter.
If the second offer was only marginally higher, you may well be able to work out a deal with the sellers. On the other hand, recognize where you're coming from: They believe (perhaps correctly) that you must sell your current home in order to purchase theirs. From your question, it's not clear that you did; that would have been spectacular timing to have received a contract on your property the day before the contingency expired. So, if you haven't sold your home, and your ability to purchase theirs depends on your sale, then they have a legitimate concern over whether you'll be able to purchase. (And, a piece of advice: In this market, even if you can purchase theirs, you really, really don't want to have purchased a new home while trying to sell your old one.)
But, see a lawyer.
Hope that helps.