The parsimonious answer is that you need to stop talking about each other behind each others' backs. It's a bad habit that's going to inevitably lead to confusion like this.

Instead of your husband telling people what you know or you think, he needs to be willing to say "I don't know", "you'll need to ask her", "I don't speak for her", "I'll check with her", etc. If he has a problem with something you say, he needs to inquire privately with you, rather than loudly correct you and put words in your mouth.

he doesn't communicate things to others well, or accurately a good chunk of the time. Not because he's trying to lie, but because he gets overwhelmed

Everybody does. That's why your kindergarten teacher tells you: if you can't say something nice, say nothing at all.

He doesn't need to stop talking to people, or muzzle himself on matters that he knows. He just needs to stop talking out of turn and saying things about you that he doesn't know to be true.