They tried taking turns, but Sam learned what it felt like to want the Ring, and knew he couldn’t do it again.

I just listened to the books again. They didn't really try to take turns. Sam thought frodo was dead so he took it to keep it from the orcs. And even after he gave it back he offered a few times to carry it again because frodo was so weak, but frodo wouldn't let him and definitely was freaked out a when sam asked. The book describes frodo suddenly thinking sam was an orc or a thief trying to take the ring from him.