Why does my fabric pucker every time I try bead embroidery?

The puckering is normal for what you're doing, and the fix is exactly what your friend hinted at. Plain cotton just can't hold its shape when you're sewing through it twice per bead, hundreds of times. Every stitch pulls the weave a tiny bit and it all adds up to that raisin effect. The hoop only masks it while the fabric is stretched, so it comes right back when you release it.

What works is giving the fabric something stiff to sit on. The most common setup in bead embroidery is a fusible woven or non-woven interfacing ironed onto the back, medium to heavy weight, or a piece of bead foundation stuck to the back with fabric glue. I've also used a scrap of felt or even stiff pelleted stabilizer when I didn't have the proper stuff on hand and it worked fine. The idea is that the backing takes the tension of each stitch instead of letting the fabric distort. Once you've got that underneath, you can pull your stitches snug again and the beads will sit flat and stay put, which solves your loose-bead problem too. Thread matters more than you'd think. Regular sewing thread is thin and it shreds or frays against the bead holes over a few passes, plus it has no body. A dedicated beading thread, the kind sold for loom work and bead weaving, is stronger and slightly stiffer, so your beads lock in place better. Size 11 beads ussually want a fine beading needle, which you already have, so you're set there.

A couple of smaller things that helped me: don't drag the thread across long distances between beads, because that's what really wrinkles the surface. Anchor and cut instead. And work from the center of your design outward so any slight tension settles evenly rather than bunching at one edge. So yeah, you weren't doing anything wrong. You just need the backing layer and a beefier thread and this stops being a problem.

I'd push back on the thread part of that answer. I've done plenty of bead work on fabric with plain polyester sewing thread, just doubled and run through beeswax, and it holds up fine. The whole 'regular thread has no body' thing sounds like something someone read somewhere. The real problem I've seen is people skipping the wax, so the thread frays where it rubs the bead holes. Try that before spending money on specialty spools and see if it doesn't fix the flopping beads just as well as the fancy stuff.

One thing I'd add is that a little give in the thread matters as much as snug tension, since cotton relaxes after handling and a stitch pulled dead tight will drag the surface in once the fabric settles. I stop each stitch with just a hair of slack so the bead sits firm but can still roll between my fingers on the fabric. Also, once a piece is finished, I baste it face down to a second backing and ladder stitch the edges closed, which traps any residual waviness between the layers where it never shows. Funny how the puckering always looks worse under a raking light than straight on.

With the person who said wax is the real fix, doubled thread run through beeswax has been my go-to on fabric for years and it never frays. One thing I do differently that made a big difference: I mount the backing and the top fabric in the hoop togehter as a sandwich, so the stabilizer stays tensioned the whole time I'm working instead of flopping around loose behind the cotton. And if a finished piece still has a bit of wave, I lay it face down on a thick towel and press it gently with a warm iron. Between that and the waxed thread the beads sit flat and stay put for me.