I'd clean and sand them, then paint them white. This sounds dated, but I do think it would be pretty: You can paint, then rough it up a bit to look like weathered white wood.

Either one. White painted wood counts as white furniture, not wood. So it matches with color and just regular wood.
With blue walls you pretty much have a blank slate. I would go with the regular color rules and start with an inspiration item that's in the room that has more than 3 colors in it. Piece of art on the wall, painting, or rug. Pick one of those colors.

If probably clean them really well then put them in place for a couple weeks to decide if I like the wood color enough to stain them or if I wanted to paint another color. I'd get used to them and then forget about staining or just decide the scratches add charter and leave them.

I would paint them bright yellow.

We have painted white wood bedroom furniture. I thrifted two dressers to use as nightstands. They are definitely in need of refinishing: do I stain them or paint them? They are currently a honey pine color but scratched and water damaged on top.

What would you do with them given the bed and other, bigger dressers in the room are white? The walls are pale blue.