But, you’re not forced to log in before you can see the prices online, meaning they aren’t changing the price based on who is logged in.

They are. Even when blocking cookies, which most people don't do or rely on the "helpful popups" that pretend to decline them, they can still fingerprint your browser, your computer, your IP. They grab your WebGL context and profile it until they identify enough unique details about your hardware and how it works. They don't need you to login to know who you probably are, where you're coming from, and what price they should show you, and if they're wrong, they can be probably entirely consistent and close enough to being accurate that it doesn't matter. If you're careful and sneaky, you can sometimes see them do this yourself using VPNs and privacy-focused browsers and addons and other tools and by seeding your cookies and ad-data to make them fit you into a different profile, but it's rarely worth the effort and it's rarely a productive use of anyone's time, since they'll figure out the real situation when you go to pay anyway. Even if they do honor the price they initially offered you and don't mysteriously error or decline or cancel the order, you didn't really hide anything from them in the end, you just coaxed a bit of money out of them, they'll eat the loss and pay for the difference easily with the thousands or millions of others who didn't notice that their price was quietly jacked up.

I sincerely doubt they’d change prices on the shelves while the store is open, where shoppers could pick an item, and then the price is higher when they get to the check out.

Oh, my sweet summer child, they are doing worse than that. They are tracking how desperate or disinterested you seem to be about prices based on past behavior and even while you are in the store. They will know who you are, they will know your circumstances, and the price will change just for you and you'd better believe they'll know exactly where you are at all times, so they'll have systems that can charge you the same price at checkout as they showed for you when you're in the aisle... if they choose to, or maybe they'll choose to raise the price at the checkout, instead... why not? They'll probably profile you well enough to guess if you'll even notice the raised price, or if you do, whether you'll actually be willing to make a stink about it. Will they be wrong sometimes? Sure it will! Do they care? You know they don't!

Of course the really savvy shoppers may be able to dodge all these tricks, maybe even benefit from some of them. But that's not something that actually bothers them. All they've got to do is do it subtly and gradually enough without creating a inescapable PR shitstorm in the process so that other, slower-to-adopt retailers don't eat their lunch while they rake in the increased profits, ultimately forcing other retailers to follow their lead and level the playing field by doing the same thing (because do you really think people are going to change their buying habits significantly when all the retailers are doing it to approximately the same degree?) instead of letting the helpless peasants-- *ahem* customers force them to back down.

Capitalism dystopia is here, don't fool yourself into believing that they won't do it, because not only will they fucking do it, they're not even bothering to hide it anymore. They aren't ashamed of it, and they aren't scared of consequences. There aren't any. They know there's probably little to nothing we can do to stop them, and they know that if we do stop them on this, they've already got a dozen other strategies in their pocket to try next, and they'll never stop. There is no consumer protection anymore. Consumers don't even protect themselves. Every industry is made up by colluding oligopolies or monopolies. They've won, they've fortified themselves, and now they're sucking the economy dry, and when they have drained all your wealth and possessions into shriveled, withered husks, if there is still work left to do, they will make you do it to survive. If there isn't work for you to do, then tough luck, they won't care if you are homeless or dying. That's... probably sort of the point, really.