Hi, Amelia.  I have wavy hair, and I french braid it at night when it's dry.  It doesn't matter to me then how it looks, but it works for me.  Wavy/curly hair is simply a bit harder to handle than straight.  I suggest braiding when the hair is damp, but not wet.  It is true wet hair is fragile, so you do need to be careful-but if you want healthy long hair it requires gentle care anyway.
Something else, your hand position when braiding affects the outcome.  I don't know how you do it now, but try keeping your palms close to your hair and see if it makes a difference.
Good Luck!
Sakina

Wet hair is very fragile as it stretches easily and can break.  If you have very thick, curly hair, I imagine it could be braided when wet if you are very careful to not pull on the hair. 

Most of us will braid our hair when it is damp.  This gives us more control over the hair as we braid; the braid will stay neater longer; and the waves created when it dries are delightful!

It is best to just put some hair cream, leave-in conditioner, or a little bit of oils in your hair before braiding it rather than braiding your hair while it is damp.

Sometimes I like to twist my hair up in a bun while it has some leave-in conditioner in it, so when I take it up it out it will have some great body and waves to it. I will even put some holding hairspray on the twisted hair bun before I untwist it to give a longer hold and better shape. Naturally, I have curly hair, but I chemically straighten it. But I am going to let it grow back out naturally.

Cristophe Soft & Silky Hydrating Curl Spray, Nuetrogena Triple Moisture Silk Touch Leave-in Cream, or Nuetrogena Triple Moisture Healing Shine Serum would all work great to help tame the curls while braiding.

If you want something organic, try Organic Root Stimulator Olive Oil cream is good too.