How do I keep my paper feathers from looking stiff?

Yeah, the flat stacked look is almost always a paper weight problem combined with shaping too late. Pencil curling does basically nothing on heavy cardstock, you need actual memory in the paper. I build a lot of birds and I've moved to 80 to 100 gsm paper for the top feather layers, basically printer paper or slightly heavier, with cardstock only underneath as the structural base. Thin paper holds a curl with just a scissor blade. You run the feather between your thumb and the flat of the scissor blade like you're curling a ribbon, one pass, maybe two, and it takes that soft cup shape that makes overlapping edges read as plumage instead of shingles. If you want a sharper curl on a longer feather, wrap it around a knitting needle or the barrel of a pen and hold it a few seconds.

Scoring also works but only on larger feathers. A light score down the center quill line lets the two halves cup away from each other naturally, which looks great on wing coverts. On tiny head feathers scoring just tears the paper, skip it there.

On the glue, ditch the stick entirely. A glue stick gives you that grab then slip behavior overnight, which is exactly the lifting you're seeing. Thinned white glue is what most paper sculptors use, roughly half glue half water, applied with a brush in a very thin coat. It does wet the paper and it can ripple if you flood it, so thin coat, press down, and it dries rigid and basically invisible. For the tiny head pieces I sometimes skip the glue bath and just touch a pin head amount of undiluted white glue to the base of the feather with a toothpick, then pin it in place with a straight pin pushed into a foam board or cork base til it sets. That stops the overnight lifting completely. One more thing on layering. Work from the tail end forward and from the bottom row up, so each new row's edge sits over the top of the row below it, like actual feathers. If you've been working head down that's part of why it looks stacked, the overlaps are going the wrong way visually.

Been making these since before glue sticks were even a thing, and the old trick we used for paper flower petals works here too. Dampen the back of the feather lightly with a sponge, then shape it and let it dry on a curved form, like the bowl of a spoon. That sets a permanent cup into the paper, no fighting the cardstock. Old florists did this with crepe paper for decades. Curious what you're using for a head form though, that kingfisher beak is a whole seprate battle.

For me the game changer on tiny pieces was a pair of reverse-action tweezers, the kind that grip when you squeeze and hold on thier own. I was losing my mind trying to press little head feathers down with a fingertip while the glue grabbed, and the feathers just stuck to my fingers and curled up anyway. With the tweezers I can set a piece and walk away from it. Mine came from a hobby shop years ago, cheap ones work fine. Pair that with a bit of patience on the glue and the head section finally stopped fighting me.

Not a bird builder, I mostly do paper leaves for wreaths, but the layering order thing clicked for me because I made the exact same mistake with leaf rows last autumn. One small thing that helped mine look less uniform was cutting each feather from its own traced template with tiny variations instead of stacking copies of one shape. Real plumage is a bit uneven. Also still figuring this forum out, is there a finished projects thread here? I'd love to see how the kingfisher turns out once the wing section behaves.

Update: tried the damp sponge and spoon trick from the second reply and the back layers finally have some shape to them, huge difference. Still fighting the lifting edges near the head though, glue stick just is not holding the tiny pieces down overnight. Is there a glue that grabs fast enough that I am not sitting there pressing each one for a minute?