Is there a trick to keeping crocheted bag straps from stretching out?

The cord core is the right idea and honestly easier than it sounds. I ran a length of cotton clothesline through the straps on a market bag I made a couple summers ago and the stretch basially disappeared. What I did was crochet the strap as a tube, roughly like you'd do tiny i-cord, then feed the cord up through the middle with a blunt needle or a crochet hook once the tube was done. Working it in as you go is posible but fiddly, whereas threading it in after means you can adjust the cord length first and make sure it sits evenly with no bunching. I tacked both ends of the cord down with a few wraps of the working yarn when I joined the strap to the bag, so it can't slide or shift around inside. If your strap is a flat row of single crochet rather than a tube, you can still do a version of this: crochet over the cord for the whole length, treating it like a core you're wrapping. Just keep pushing the cord along as you work so it never snags or gets caught in a stitch. A tighter stitch like slip stitch all the way down will help some, but on its own it still stretches, just less. Slip stitch straps also tend to be narrow and stiff rather than truly strong, and cotton worsted will give under a grocery load no matter how tight you grip the hook. The cord is what actually carries the load. I'd say the attachment points are the other thing worth checking, maybe even the bigger culprit. If the handles are attached at just a couple of stitches on each end, all that weight concentrates on a tiny area and they'll stretch or tear away from the body no matter how solid the strap itself is. Spread the join across a wider section, like four or five inches of the bag's top edge, and work through both the handle end and the bag body with multiple passes. Sewing the ends down with a yarn needle over a decent span makes a huge difference. One more thing: a strap that runs continuously over the top of the bag, stitched right into the body along its whole length instead of just at the two ends, distributes the weight way better than two seperate tabs.

One thing that helped me was realizing my starting chain was the stretchy part. I used to chain a long length and work single crochet back along it for handles, and that little chain just keeps giving no matter how tight your tension is. Switched to foundation single crochet for the strap and the difference was night and day. Held double with a smaller hook on top of that and my grocery tote hasn't budged in over a year. Cotton also does that thing where once it's stretched wet and heavy it stays stretched, so I stopped washing mine hanging full of produce too.

Been fixing floppy bag straps since the 80s and the single pass of sc was never going to hold, no matter how tight you work it. My old trick is to work the strap twice, up one side and back down the other, so you end up with a doubled strap like the old belt-style handles we used to make before everybody switched to webbing. Doubled fabric fights itself, that's half the battle. Also don't store the finished bag hanging by its handles, cotton will creep under its own weight over months and you'll be back where you started.

Mostly a knitter here so I might be out of my depth, but when I made a bag for hauling library books I swapped cotton for a linen blend on the second attempt and the straps barely budged after that. Linen has almost no give compared to cotton, so it might be worth thinking about yarn choice too and not just construction. Curious wether that cord-in-the-tube trick would work on a knitted strap the same way, since mine are I-cord style and would prolly swallow a cord nicely.

Something I learned the hard way with a cotton bag of my own: cotton keeps creeping long after the first few uses, so a strap that seems fine on day three can still be longer by the end of the month. Now I torture-test straps before attaching them, hooking the finished piece over a door handle and hanging a bag of books off it for a day or two, which forces whatever stretch is left to happen while I can still undo it. If it grows visibly I rip back and rework rather than trying to compensate with tighter tension, because tension alone never survives a real grocery run.