Some Legal Advice...
Nothing they said was true. That being the case, I would go in there a cause a scene so that they feel embarrased in front of other customers. Then maybe they'll do what they should have done (give you a replacement or a refund) just to keep you from causing a scene and making them look bad.
Also, I would mention to them that any product you buy from a store comes with an "implied warranty of merchantability", meaning the item must be fit for the purpose for which it was intended. Since your watch will not run for more than a few hours without stopping, it is not fit for the purpose of keeping time, so the "implied warranty of merchantability" has been breached. Therefore, the store MUST either replace the watch or give you your money back. It's that simple. If you go in there sounding like a lawyer who knows what he's talking about, they will be more likely to get you off their back rather than deal with any legal headaches you might initiate.
Also, do NOT let them send the watch to New York or anywhere else as you may end up voiding the Omega factory warranty if they screw it up. I just don't trust these guys.