Just to be clear as I did a lot of Clothes Dyeing in the
Sewing Machine before I finally got the 6 colors figured out for the Dyeing Pots:
Tailoring a Dyeable Clothing item in the Sewing Machine with the Prismatic Shard or Rainbow Shell will get you the choice of color (with color sliders) of that one item, which I didn't know until reading
@imnvs post above (just tried it, that's too cool and easy).
Tailoring a Dyeable Clothing item with any Dye in the Sewing Machine will get you about a 1/3 color conversion towards that color, usually ending up in a dull gray. Doing the same again with the same color gets you a faded color version of that color, doing the same a third time gets you the saturated color version of that item. This is expensive depending on your Dye source but is what I did for quite a while.
Using the Dyeing Pots instead requires 6 dyes of the different colors (still somewhat expensive, depending) but allows you to color match 2 the Dyeable Clothing items you're wearing and gives you the color sliders for each item so you can preview it on the Dyeing UI.
Tip: Put your UI to max size beforehand to try to see the color popup easier as it's just too small, this interface needs to be bigger.