When it comes to anxiety, I've found that the truth is sometimes the easiest.

You could simply tell anyone that asks that you're dealing with some anxiety issues and have made alternate arrangements with professors in regards to public speaking.

It's simple, to the point, honest (so there's no lie to remember) and doesn't leave any room for discussion. I know it's hard to reveal something so personal...I've only just now started to be able to do so. I have practiced telling people by telling my dogs, or a mirror, or just saying the words out loud. Being able to honestly tell people that you're struggling with anxiety seems to have eased the terror around telling people something so personal. Each time I tell someone, it gets a little easier, a little less scary, a little less painful.