Merhaba.

I need to have this one translated:

I have 3 last exams left next week and after that I will have free time.

My try:

Sonraki hafta sadece 3 son sınavım kalıyor sonra boş zamanım olacak 

Perhaps my version my be ok, but is it the most natural way to express this?

Or would a native say it in a different way?

Sonraki hafta ancak 3 son sınavım var, sonra boş zamanım olacak.

i am not a native speaker, but this is my try as learner.

your version is mostly OK, but it should be "son 3" not "3 son".

Better to say: Gelecek haftaya sadece son 3 sınavım kalıyor, sonra boş zamanım olacak

üç sınav (three exams)

son üç sınav (last three exams)

Except these adjectival syntax, you have no problem. But I´d like to offer more professional way:

"gelecek" is better than "sonraki" there. Both are the same, but we have gotten used to usually use it. Otherwise, yours one is correct, too.

By the way, it would be nice if you don´t omit the comma when you separate the sentences.

gelecek hafta+a(dative)--------->gelecek haftaya (to the next week)

haftaya: 1 to the next week, 2 next week.

What´s more, we usually don´t use the adjective "gelecek" when we use "haftaya".

Haftaya son üç sınavım kaldı, sonra boş zamanım olacak. 

(I have last three exams remained to the next week, then I´ll have free time.)

People understand that three exams of you remained and those will be taken next week.

Maybe he means "final" with the last exam. I have 3 finals left. 3 finalim kaldı.