We're usually taught (in Korea) to say getting old to describe that something/someone is aging, but I found out this can sound unnecessarily exaggerated as if it's running out of its lifespan. To tone down this expression I was thinking about saying getting ages instead of getting old, but I'm not sure it is idiomatically natural. I cannot easily find the usage of this, so I postulate maybe it's not that common and there must be a better expression.
No, "getting ages" is completely wrong here. If you were asking a group of people to tell you their ages so you could write them down - for example, if you were conducting a survey - and someone asked you what you were doing, you might casually say:
What are you doing?
I'm getting ages.
This is totally different than what you're tying to say.
There is nothing wrong with "getting old." You can weaken it by saying "getting older." If you are a child or young adult, you can say "I'm growing up" or "I'm maturing" (the latter has some additional connotations you may not want to evoke).