Why does my lettering pucker when I stitch on striped fabric?

The gauze-y weave is most of your problem, not your tension. Loose open cotton needs something behind it or the stitches just sink and wander wherever they want. My go-to is a piece of wash-away stabilizer on the back, or if you don't have any, a scrap of lightweight cotton ironed behind the hoop area works too. Some people hoop a layer of tulle with the fabric and stitch through both, then dissolve or cut the tulle away after. On that kind of fabric I also loosen the hoop a touch, snug but not drum-tight, because pulling it super tight just stretches the weave and then everything puckers when you take it out of the hoop. Stitch tension matters too. On gauzy fabric, if you pull each stitch hard, the letters bunch like you're describing. Aim for the thread to lie on top of the fabric, barely denting it. Back stitch or stem stitch with two strands usally behaves better than three on loose weaves, so your scrap experiment was pointing the right direction. And yes, absolutely use a water soluble pen. It's fine on cotton, that's exactly what it's made for. Test it on a corner of the actual project fabric first just to be safe, but I've never had one bleed or stain. Draw your baseline and letter height marks, then ignore the stripes completly while stitching. That drifting sideways thing is real though. Strong stripes mess with your eye, your brain keeps trying to align the letters to the stripes instead of to your guideline. Two tricks: draw the full baseline plus a top line so you've got a band to letter within, and step back every few letters to check. Also, lettering looks straighter if you lightly pencil the actual letter shapes rather than freehanding between two lines. For the phrase over the state outline, consider running the baseline parallel to the stripes rather than fighting them, since a slight angle reads as a mistake but an intentional alignment reads as a design choice. If the stripes are truly fighting you, a square of white or muslin appliqued behind the lettering area first gives you a calm background to stitch on.

Agreeing with the backing advice, it made all the difference for me too. I stitched a quote on some loose gauzy linen last year and kept fighting wavy letters untill I basted a scrap of an old pillowcase behind the hoop area. Overnight the stitches started sitting on top of the fabric like they were supposed to. I just snipped the basting out when I finished and no one can tell it was ever there.

The stripe-drift thing got me bad once on a striped tea towel project. I kept correcting against the stripes mid-letter and everthing leaned downhill. What finally fixed it was a sheet of wash-away film on top - I traced the whole phrase onto it, stitched right thru it like a pattern, then soaked it off in the sink. Watching the letters instead of the fabric while I stitched stopped the drifting completely. Might be worth trying on a scrap since you're low on fabric to practice on.

Not buying the 'I've never had one bleed or stain' part from the first reply. Those pens absolutely can leave a ghost mark, mine did on a blue cotton once, and it only showed up after the piece dried. The trick nobody troubles with is you can't iron the fabric before the marks are rinsed out, heat sets that stuff. So test it, sure, but also rinse first and press after. On a striped thrift find you'd be even more annoyed than I was, since redoing a section means matching the stripe spacing.

Update: basted a scrap of an old pillowcase behind the hoop like the second poster suggested and the letters finally sat on top instead of sinking into the weave. Used the wash-away film on top too for the phrase, traced it first and stitched through it. Even got the rinsing-before-ironing warning just in time, heat would have set any leftover marks. The stripes still fight me a little but the text actually reads straight now.