Eventually, I did find bodywashes that weren't irritating—the trick was to look for things without any or very minimal fragrance. And interestingly enough, I'm back to being a bar soap person now, but that's another story.
For the times I turn to bodywashes, though, I make sure they're fragrance-free and good for sensitive skin. Luckily, nowadays there are so many options for anyone who's avoiding fragrance. If you have a similar sensitive skin issue, or just don't like the added fragrance, take a look at some options below.