"i will write more (to you) "
1. 나는 더 쓸 거야. (your version)
2. 나는 더 쓰겠어.
What you wrote is perfectly correct - that is the most common way to talk about the future or one's will.
#1 is has a broader sense, and can also mean something set to happen in the future ("I am going to write more").
#2 means one's will (-겠어 has another unrelated usage too), so its usage is a little more limited.
See these examples:
- 쟤는 내년에 학교에 들어갈 거야 - He is going to school next year. (-겠어 is not possible).
- 나는 이 일을 꼭 해내겠어 - I will see this through no matter what. (both -겠어 and -ㄹ 거야 can be used).