Hello !
First, I found out that 지긋하다 becomes 지긋이 but not 지긋히 when it is used as an adverbe.
So you can say 나이가 지긋한 사람 or 나이를 지긋이 먹은 사람. Both of them.
We use "나이가 지긋한~” or any other applied sentences to describe elders in a polite way. We don't use it to designate young people.
And "나이를 먹다” is same as "나이가 들다” (grow old)
Example:
나이가 지긋한 노신사
A gentleman of a certain age