Have I made a huge mistake by sharing my news?
Not necessarily, as long as you handle it correctly. The rumor-mill does what it does, doubly so when you've got a new company taking over an existing one.
Unless you announced your news by something that leaves a trail (such as an e-mail, SMS message, etc.), then the rumors that you were offered a better position can't really be traced to you personally (as you say, people were already spreading manufactured news around before you even mentioned anything). So as long as you don't spill the beans about having been the one who spilled the beans, probably nothing will come of your mistake.
Or if you must discuss the issue, frame it as you have above. Your understanding was that you were to keep the matter quiet until you received the formal/written offer, and that once you had the formal offer you were free to discuss it. In other words, frame it as a miscommunication/misunderstanding as opposed to "I deliberately did something I was told not to do".
Note that depending upon your location, you may have a legal entitlement to openly discuss your terms/prospective terms of employment with your colleagues regardless of whether or not your employer wants you to. Not that it should be necessary (or wise) to invoke such an entitlement in this instance. Just keep a low profile and probably it will all blow over on its own.
Could they/ would they rescind the offer.
In general I don't believe anything would prevent them from rescinding the offer if they really wanted to. Unless there are some specific contract terms or other agreements in play in your specific circumstances. But in the general case, yes, they can rescind the offer.
Will they rescind it, however? I'd say almost certainly not. Especially if you don't make any further noise on the subject, and provided that there's no "smoking gun" showing that you clearly and deliberately ignored their request (like an e-mail showing you bragging about the offer 5 minutes after you received it).