Buyer makes an offer. Counter offer is made. Buyer has meantime seen another house they prefer, so choose to ignore the counter offer. Is the Seller then permitted to accept the original offer? Does it make any difference if the counter offer is in writing or verbal?
When they made a counter offer they essentially rejected your offer, and you are no obligation to accept theirs. Nor are you obligated to your original offer. That is the risk a person takes when they make a counter offer. Their agent/representative should have advised them of this risk.
Because time is of the essence in most of these transactions their delay with the extra round also made your original offer void. You were right to keep other properties in play.
Offers usually have an expiration deadline. As long as the signed offer is returned by the seller to the potential buyer before the expiration - it is a valid contract. The fact that the seller countered and his counter-offer wasn't accepted is irrelevant.
The buyer can void the offer, as long as it is not yet accepted, by notifying the seller in writing that the offer is null and void.
I'm not a lawyer, you should ask your real-estate attorney to be sure, but that is my understanding of the contract law.
It doesn't matter if the buyer breaches the contract during the due diligence period, because most due diligence periods allow the buyer to breach the agreement without cause. Unless there is due diligence money added to the contract (different from earnest money), then breaching the contract costs nothing to the buyer.
This happened to me. I learned the hard way: until the due diligence period is over, the buyer can renege without consequences. It's one of the risks of being a seller.