I wouldn’t say I have much experience with dip dye but I have played around with it a bit for a styled photo shoot. I dipped the paper (it was Crane’s Lettra) first and then once the sheets were air dried I stacked books on top to make sure they were flattened as much as possible. I was doing a three color dip dye and found that I was much happier with the results when I painted the layers of color, which also seemed to take about the same amount of time as dipping (painting would also lessen any curling on the sheets because you wouldn’t be saturating the paper, but on the down side you only get your dye/color on one side of the paper rather then front and back).
I then printed from a photopolymer plate on a tabletop platen press (I didn’t dampen the paper again, just printed on dry sheets and achieved a good impression). I attached a photo of the painted version that I ended up using.
Hope that is helpful!