What is the difference between 자야 돼 and 잘 거야 ?Feel free to just provide example sentences.

어/야 돼 is "have to", and 을 거야 (을 거예요 the most formal version) is "will" , a future form, es: with 어야 되다:
화장실에 가야 돼요- I have to go to the bathroom.
저는 한국에 가야 돼요. I have to go to korea.
자야 돼요. I have to slepp.

Exemple with 을 거야.
저는 잘 거야. I will sleep.
저는 내일 그 책을 읽을 거야. Tomorrow I'm reading a book.
저는 일본어도 공부할 거야. I will study also japanese.
저는 오늘 그 사람이랑 만날 거야. Today I'm meeting with that person.

(을 거야 is a informal form that you can use when you speak with your friend or parents, if you speak with a person that you see for the first time, you could use 을 거예요).