Chewy sticks from fresh wood

Hi all, I have some questions for those of you who cut your own wood for your degus to chew. Awhile back I read that mulberry wood is a safe wood for pets. It turns out that I have a mulberry tree in my yard, so I have cut some branches from it for the degus. My question is this: do you think the wood needs to cure/dry in order to be safe for the degus? Realistically, I know that in the wild they chew on fresh live wood all the time, but I want to get your input anyway in the interest of safety.

I have put the cut branches aside to dry, but we're having very wet weather here and they are drying very, very slowly.