Why is 才 correct but 还 not in 我才27岁?
Although you are trying to say "I'm still 27.", "还" does not work in this context, since it is only used to imply an action is still ongoing or something still exists. Grammatically speaking, it should be used before a verb, not a noun.

On the other hand, "才" in this context is used to emphasize a (merely) small quantity. Therefore, it should be put before a noun. There are other usage of "才", making it can be put before a verb, but that's a different case.

Maybe this example will help you to notice the usage of "还" and "才" (example from Chinese Grammar Wiki).


  你才一百斤，还要减肥？