The coin on the left on your original post is a British (Britannia) Halfpenny, minted between 1729-1759. These British coins were the main small denomination coins used in colonial America, the most common kind of colonial coin. I love finding British Halfpennies, so far my oldest British halfpenny is from 1696, minted under the reign of William III. Congrats on your great coin and I hope you find more.
Also, I noticed that Britannia looks slightly different, maybe a contemporary counterfeit (a fake coin with intent of spending, made at the same time that the coin it copied off of was being used), I'm not sure that that is what it is, for there are many die varieties and the corrosion may make Britannia look slightly different.
Definitely a Large Dandy button. I have dug a few of the same one. Nice early items you have there. If we only knew the history of our items. That would be the ultimate