How do I keep fine underglaze detail from smearing when I dip my pieces?

The blur is happening because dipping re-wets the underglaze and the liquid glaze acutally picks up a little of it as it slides around. Your lines aren't dissolving, they're getting dragged. Two things fixed this for me when I was doing fine sgraffito-style linework on porcelain beads.

First, go thicker on the underglaze. Three coats, letting each dry till it's not tacky. Thin single coats look fine before glazing but they're basically a watercolor film at that point, and one dip is enough to disturb them. Thicker strokes hold their edges a lot better.

Second, and this made the bigger difference for me: fire the piece once after painting, before you glaze. So bisque, paint your dragons and fish, then run it through another bisque-ish firing to set the design. Once the underglaze has been fired onto the surface it stops being water-soluble, and you can dip in clear glaze without the lines moving at all. It feels like an annoying extra firing, but it saves you from exactly the batch-wasting spiral you're in. If you'd rather skip the extra firing, brush the clear on instead of dipping. Brushing disturbs the surface way less than dunking, especally if you load the brush well and lay it down in one pass instead of scrubbing. Thinning the glaze and doing two or three lighter coats also works, it just takes longer and you have to watch for pinholes. On the food surface question: most commercial underglazes are fine under food-contact conditions once fired properly, but bare underglaze tends to be a bit matte and porous feeling, which is annoying for something that will get touched and handled constantly. A clear coat makes it glossy and way easier to wipe clean. For the inside where a cookie sits, I'd definitely glaze it. If you're worried, check the label on your specific underglaze line, since some pigments aren't rated for food surfaces on their own.

Another way the old studio folks handled this: skip the underglaze-on-bisque order entirely and go majolica style. Dip your bisque in clear first, let it dry, then paint the dragons right on top of the raw glaze with stains or underglaze. One firing instead of two, and the lines stay razor sharp because nothing's running over them. That's how the Italian earthenware decorators did it for centuries. Catches some folks out because you have to handle the glazed piece gently and never overlap a stroke, but for fine linework it's hard to beat. Worth a test tile before you commit a whole cookie batch.

I had the same line problem trying to do tiny patterns on a batch of tea saucers. The biggest change I made was being way more aggressive with the drying stage after the blue underglaze was on. I used a small fan to really dry it out until it felt bone dry, not just tacky. The difference was huge. When it’s totally dry, the pigment seems to lock in its edges way better before I go in with the clear glaze. Also, since you're doing porcelain, are you finding the glaze pulls differently on the curves versus the flat areas?

Update: tried the fan thing first since it was easiest. Dried the underglaze till it was bone dry and the lines did hold up way better in the dip, still a tiny bit of softening on the thinnest strokes. That majolica order from the second reply is tempting but I'm confused about one thing. If I paint on top of raw glaze, won't the brush drag the glaze itself and mess up the dip coat?