FYI, there was just a big BeckHook Pride Festival this past weekend.
https://www.thedailycatch.org/articles/beckhook-pride-the-parade-and-celebration-in-photos/
I grew up in that area and my parents still live there. Rhinebeck is a bit more "fancy," with more upscale restaurants and shops. Red Hook is a bit more down to earth but has a great burrito place called Bubby's. You will go to the grocery store and find a lot of quiet conservative people like my parents, but there will also many college students around wearing creative outfits. You do have Trump signs here and there, and also my favorite: the huge red pickup truck with the bumper sticker that says: "Protect the Boarder!!!"
I haven't spent a lot of time across the river or north and south recently. I've heard Hudson is great but I haven't been there in decades. I also don't have a dog so I can't speak to how dog friendly places are. Good luck with your search...there's nothing like the Hudson Valley in the Fall when all the leaves are changing!
P.S. Want to buy my parents' house? It needs a lot of work (okay a total reno) but is close to the village. LOL.