还 has the meaning such as "still" or "yet".
Why is 还 not correct in this situation?
- For 还 to mean "still" you need to use it before a verb like 是 (is / be)
Example: "我的生日是明天，所以凌晨前我還是 (is still) 二十七歲"
- For 还 to mean "yet" you need to use it with 未 and before a verb
Example: "我还未到二十七歲" , "我还未滿廿七歲" or "我还未足二十七歲" (all mean I am not 27 yet)
"我才廿七歲" = "I am only 27" implies it is a young age from your point of view. It is in fact short for "我才(只有-only)二十七歲"
The simplest answer to the question "你多大年纪?" is "我今年二十七歲" or simply "我廿七歲" This answer already indicated you are currently 27, which mean you are still 27, not 28 yet
Edit:
I was going to imply is not if I'm (still) 27 or 28 but if I'm also as young as the friend.
It would be "我也祇是二十七歲"
[我也 (I too)] [祇是 (is only)] [二十七歲 (27 years old)]
No matter your friend is 21 years or 30 years young, the above statement implies you are also young at the age of 27