Standart contract? Far or Far/bar was used?
if an offer is not signed by buyer is not an offer.
If the offer is signed by buyer and seller is a contract.
If there is a counter offer from seller (still an offer and you have 48 hrs to response the counter offer)
At the time the buyer present the offer there is a period of time you should put in the blank space, if there is not a date or specifict amount of days, they can go back to you after few months and response your offer.
Fi after months they have missing signatures and one of the parties wants to walk away is free to do so, Since you are right there is not contract.
But you should contact your lawyer and find out a legal advise prior to continue the dispute.