Is it okay to say, "I am increasingly older." when I mean to say "I am growing more and more older."
No, the sentence is not grammatical as it stands - or at least, it does not convey the meaning you intend. Semantically, it seems confusing, at least.
I am older.
This means that you are comparing your age to someone or something else.
I am older than my brother.
My brother is 34, I am older.
The construction of an attributive with the copulative am means that you are describing a state, not a process. (*)
Adding an adverb (increasingly) to it does not change that. The adjective may give the idea that you describe the process (getting older), but the verb am describes that it is a specific momentary state (am older). You can mix those up, but the meaning will not be clear.
The verb get is indeed a better choice:
I am getting increasingly older.
This sentence makes perfect sense. But it is, as rogermue mentioned in his comment, a so-called platitude. Nobody ages "decreasingly", so it seems a bit meaningless to add that your getting older is happening in an "increasing" way.
(*) Yes, if you use the correct attributive, you can use am when describing a process; you basically describe that your state is "being in the process":
I am ageing.