(I did do a search and saw that someone asked a similar question a few years ago, but the answer only touched on music and wasn't hugely specific, so was hoping for a more in-depth one)
This is fun and falls way out of my typical area of expertise, but I happen to have a decent amount of work in the area. The closest group to the Amish that I can talk about is the Puritans of New England. Now, by the turn of the century in 1700 the Puritans had largely marched themselves off the pages of history, having largely imploded as a society. But, having entered their death throes in 1692-1693 they are close enough to your timeline. However, the Puritan name has taken on some meanings that don't give us a clear idea of who they were as a people. We often use Puritan or Puritanical to describe people who are prudish or buttoned up, but that is actually more attributable to their territorial neighbors the Pilgrims. Why do I tell you this? Well it is to say that the Puritans did believe in partying or reveling, but in moderation. Drinking was an acceptable activity to engage in within reason, so long as you were not drinking health or what we may call now "toasting." Sports and vigorous activity were generally frowned upon and the Puritans were even furious that King James legalized the playing of tennis on Sundays. A quotation from Peter Wagner's work on the matter is especially illuminating as far as physical activity goes: "The Puritan, who held the concept that work and piety were synonymous, denounced any form of play as evil. Whatever was pleasurable had to be sinful." ^(1) So that sort of puts dancing out of the question. So what did they do for fun exactly and what might a Puritan party look like?
It was mentioned previously that alcohol was permissible, it was not only permissible but even church sanctioned at times. One church parish in early New England lists "malt and wine" among its lists of poor relief.^(2) Inns that served booze were also a fairly common and legitimate endeavor for Puritan people, often being the only places women could be permissibly employed. This sort of activity was justified in part by the Puritan ideas of their own theology: "God has given us temporals to enjoy," say Minister Joshua Moody, "we should therefore suck the sweet of them, and so slake our thirst with them, as not to be insatiably craving after more."^(3) What the good minister is trying to say here is that things enjoyed in moderation will prevent the desire to over indulge in them. So if you and I get together at the inn and knock back a few pints of cider without getting falling over drunk, all is well with the world. The Puritans loved things that were orderly yet pleasurable. Smoking was also something that was fair and pleasurable for the Puritans. The tobacco rich colony of Virginia to their south provided them with a steady stream of tobacco to enjoy. This again was something to be enjoyed in moderation, as Puritans viewed any distraction from economic prosperity and the contemplation of God to be inherently evil.
Some physical activity was also permitted, going back to order and moderation activities such as fishing and hunting were fine by Puritan standards. Perhaps more related to your question of partying would be shovel-board and bowling. Early forms of what we may call shuffle board and indeed bowling. This early form of bowling as our modern day bowling were not physically strenuous by most standards. It consisted of rolling a wooden ball of sorts at pins and trying to knock them down.^(4) Generally it was played out on a lawn of a home or inn. So sport wasn't outright banned or seen as a deeply sinful act. Though in 1647 the legislature did attempt to ban the sports as they saw people taking them to excess.^(5) Gambling on these sports, however, was strictly forbidden. Interestingly enough, horse racing was not banned either. Though it would not have looked like the races you might see at the Kentucky derby. They were more akin to the "wholesome trotting races" alluded to in The Music Man. Vigorous racing would have been totally out of the question. Speaking of vigorous physical activity, let us broach a subject well known to us all: sex. Puritans did not deny the sexual nature of human beings, in fact prominent Puritan minister John Cotton fully acknowledged "...I must be pardonend in it, if I can account it no other than an effort of blind zeal, for they are the dictates of a blind mind they follow therein, and not of that Holy Spirit which saith It is not good that man should be alone."^(6) If you were a married person you were encouraged, nay, mandated by heaven to go forth and multiply. The Puritans took this to heart and as a result churned out massive families of sometimes up to a dozen children. Again, they saw marriage as a way to moderate the greater sin. If you were having sex with your wife a lot, that meant that you weren't doing so with your maid or slaves.
With these examples established, and an allowance that perhaps you and I are not the purest of Puritans, let me paint a narrative picture of what partying "like it's 1699" might look like to early New Englanders: You and I decide to celebrate a massive business deal in which we have both received contracts for new ships commissioned by some wealthy Virginian planters. We take a leisurely stroll in fine fitted clothing down to the track to see a horse race being put on. There we make merry by commiserating with our fellow business men and chat about the goings on about town. After we have seen our fill of racing we walk down to the inn where we inevitably end up downing a few pints of cider or beer. Feeling red in the cheeks we head outside to play a rousing game of bowling on the lawn. Feeling full of this merriment we retire to our respective homes for the night. We take a moment to say our prayers, tuck the kiddies into bed, and wile away the night hours having sex with our wives.
This may seem tame to you and I, but running all of those together in one night would constitute taking things to excess for the Puritans. Partying often implies a degree of excess, however. So there you have it, a party in the Massachusetts of 1699.
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Not an answer, but you mention an older question & answer. Would you mind linking that for us?