1) You get used to it. Eventually you can come to prefer cheap whole grain bread vs cheap white bread.
2) Try non-wheat breads like dark rye. Dense dark ryes like a good seeded deli rye or even a cheap pumpernickel remain moist and chewy without the added sugar, which is why wheat breads tend to have HFCS or honey added. Note that lots of mass market rye breads contain wheat flour, so be careful about making assumptions and generalizing based on a few samples here or there. With wheat flour it will become stale quicker, and there'll be a less complex flavor profile.
3) Try higher quality breads. Note that higher cost does not necessarily imply higher quality, though that's more often true than not at supermarkets. Basically, the point is to get more flavor with minimal cost in carbs and calories. So a sprouted wheat or bread with nuts might help.
4) Maybe you're just a picky eater, which is a real thing. If all you like are, e.g., pizza and french fries and similar foods from childhood, and especially if things like the _texture_ of other common foods are offensive, it might be a psychological thing. Most people have psychological barriers to eating and enjoying different foods. It took me years to learn to tolerate Japanese cuisine--I could eat sashimi, no problem, but the flavors of sushi and Japanese cuisine in general were off-putting, much more so than other cuisines, even ones that weren't to my tastes. With _effort_ I learned to enjoy some of it. A simpler example is ginger--I hated ginger until I didn't. But some people are at the extreme end of the scale and it's much more difficult to learn to enjoy something even with effort. In retrospect, I've probably known several legitimately picky eaters. It's not uncommon AFAIU and it's fair to dial back expectations if that's the case. Indeed, foods with more complex flavors and textures as I recommended above might be overstimulating for picky eaters.