Start by repeating everything your wife says to you, back to her to make sure you understand and have "heard" what she's said. Then maybe give a rundown of what you are planning to do to get it in your head too. Work to not go off script?

Why not suggest some marriage counseling and the two of you work on your communication with each other? I don't think you should be made to feel like a bull in a china shop, especially if you are well-meaning which it sounds like you are. Everyone makes mistakes but you shouldn't have to do anyone's discussing for them, especially with your in-law's, that's your wife's problem and if she doesn't like how you handle it then she has to handle it herself; no fair her being armchair quarterback and saying what you can and cannot say! She only has control over what she wants to say. I think it is too bad of her/others to put it all on you that you are the miscommunicator and problem maker.

Sincerely apologize to your wife, tell her how you feel and learn to zip your mouth about other people's problems. Practice listening instead of doing. Pretend you are strapped to a chair and your mouth can only repeat what is being said to you to make sure you understand what is being said, not to try to "fix" it (if it is someone else's problem) or even advise on it if you don't have your own similar problem and can tell your own story of how you dealt with it using "I" words (instead of "you should do. . . .")