Is it normal for embroidery this dense to pucker after washing?

Yeah, skipping the pre-wash is almost certainly part of it. Old embroidered shawls are notorious for this because the stitching and the ground fabric shrink at different rates. Dense embroidery acts like a patch of stiff, tight fabric sitting on a loose weave, and the first wash is when the two settle against each other. Once that's happened, the puckers around the paisley aren't going to fully relax, though you can improve them some.

What I'd try now: press from the wrong side with a pressing cloth, lots of steam, and press, don't iron. Lift and set the iron down, because dragging it over embroidery flattens the stitches and can distort them worse. While each section is still warm and damp from the steam, mold the puckers flat with your fingers and hold it there until it cools. A clapper or even the back of a wooden spoon helps. You won't get it perfect, but dense embroidery areas usually never sit totally flat anyway. Some of that texture reads as part of the look once you stop fighting it. For the next one, yes, wash the shawl first, even twice if it's a cotton blend. Then cut with the embroidery in mind. If the border crosses where a shoulder seam would go, shift the pattern piece so the seam sits in a plainer area, or accept that the seam will be bumpy and topstitch it down deliberately so it looks like a design choice instead of an accident. Staystitching the edges right after cutting also keeps a loose weave from stretching out before you sew it.

On underlining: a light cotton batiste or silk organza behind the body pieces will calm down the puckering a lot and stop the loose weave from pulling out of shape at the seams. It does add a little body, but for a short-sleeved shirt it usually still drapes fine, especially if you only underline the main panels and leave the sleeves alone. Silk organza is nearly weightless, so that's the one I'd reach for before anything heavier.

One more small thing: next time hand wash or use a mesh bag even on gentle. The agitation is what really torques embroidered areas out of shape.

I made a vest out of an embroidered bedcover a couple years back and had the same warping along every seam that crossed a motif. What actually fixed it for me was hand basting a row on each side of the seam line before I stitched, right through the thick parts. It holds the ground fabric and the stitching togehter so the feed dogs can't shove them at diferent speeds. Slow going, maybe twenty minutes per seam, but those seams stopped rippling entirely. I still baste anything dense like that by hand now, machines just don't handle the thickness change well no matter what foot you use.

Agree with the organza suggestion, it really is light enough that you forget it's there. I underlined a tablecloth blouse with it last year and the drape stayed soft, only difference I noticed was the seams stopped rippling after washing.

One trick I picked up with that same project: press the embroidered spots face down on a thick fluffy towel. The pile lets the stitches sink in instead of getting crushed, and the puckers flatten out better than they do on a flat board.

With mine I got in the habit of spritzing the whole textile with water and letting it dry before I ever put a pin in it. Even if I don't fully wash it, that gets the ground fabric to move once so it's mostly done shifting by the time I cut. Worked on a crewel curtain panel I turned into a jacket, barely any rippling after that. Also for the shirt you alredy made, if the shoulder puckers won't press out, I've had luck stitching right along the seam line through everything after the fact. It kind of tacks the embroidery down where it wants to bunch and makes it read as intentional, sort of a quilted look.

Update: I tried the spritz-and-dry trick on a scrap of leftover shawl fabric and wow, it really does shift on the first wetting. That confirms my skipping the pre-wash was the big mistake. Also tried pressing the puckered shoulder seams face-down on a towel like someone suggested and it flattened them out a lot, not perfect but wearable. Question though: does the organza underlining change how the collar sits, since mine has that embroidered border running right through it?