I didn't stabilize, I only dyed woods, mostly maple and birch. I turned to shape and then applied the dyes. I normally did not use pressure or vacuum, just slap on the dye in what ever pattern I decided I wanted. Sometimes I dipped, sometimes I brushed they dye on, other times I sprayed the dye. Sometimes I turned it "large", then dyed with a dark color and then finished turning and dyed with a different color. Sometimes I brushed on a darker color in what I called a zebra pattern, then sprayed on a lighter color. Sometimes I brushed on a sanding sealer in a zebra pattern, and then applied the dye.
In other words there are no real rules on how you have to use dye. My dye of choice was Fiebings alcohol based leather dyes, they have a rainbow of colors and a million more choices. My other dye was Vita Critter Bird Toy dye, limited colors, but vibrant colors.