On re-reading my last post I don't think I was clear about the pins & tails positions. The tails should be cut into the waterfall (the vertical boards) and the pins cut into the top (tabletop). I understand that's what you wanted anyway. I'll try and explain why. Your tabletop is longer than the sides, so it has more power in its ability to cup than the waterfall ends. The dovetail tails will encapsulate the pins and limit the cupping movement.
However, if you put the tails on the tabletop, there is a possibility they could pull out from the pins. Although with today's modern PVA adhesives (and I would use Titebond II here for the longer open time), the glue joint will actually be stronger than the wood.
I would be strongly tempted to make the tails and pins equal in size, not like a London dovetail with their skinny pins and wide tails, but more like a traditional European dovetail. The reason for this is that while the London style looks pretty it's probably not strong enough here, and the pins may break.
There is more power in wood movement than you can imagine, and I've seen wood (pine) break apart welded steel frames when it's expanded. Anyway, don't get too hung up on the moisture content as that's not a big issue here. You're working with one board, its moisture content is going to be reasonably similar throughout its length and it will expand and contract across the width roughly equally everywhere.
The biggest issue is the way it's been cut, your board is flat sawn and goes very close to the centre. If you look at the rings on the end you'll see what I mean. Oak does have a tendency to split down the middle as the tension forces at the edges pull on the centre. You'll probably see small cracks appear all over the board in time, they're thin but you can see and feel them. It is part of the character of oak. You can learn to work with them and live with it, or you can fill them with resin, or you can rip the board down and joint all the edges (lose about 10% of the width and 90% of the character in the process) and glue it all back together.
Whether you leave the pins protruding slightly, in an Arts & Crafts style, or plane them flush is up to your design aesthetic. Personally, for a waterfall style end, I would flush them, maybe even round the corner over and I would definitely mitre the end 1/2 pins so you have a mitre joint on the edge.
Whatever you do finally end up doing, I'd love to see the finished project. I'm sure it will look awesome.
Take care and stay safe,