Is this correct?

I am not Irish, but my heart and soul are Irish == Ní Gael mé, ach tá mo chroi agus m'anam Éireannach

Any corrections and/or alternate suggestions are welcome as always.

I wouldn't alternate between Gael and Éireannach. Both mean Irish, but Gael implies "I am Gaelic" to my mind. Éireannach is more inclusive.

Ní Éireannach mise/mé, ach is Éireannaigh íad mo chroí is m'anam.

Or

Ní Gael mise/mé, ach is Gaeil íad mo chroí is m'anam.

I think, in the context, the latter may be better, but wait for more suggestions 7 corrections.

Ní Éireannach mé ach tá mo chroí agus m'anam gaelach.

It doesn't natural to me to use Éireannach with heart/soul.

Ade, Éireannach/Éireannaigh and Gael/Gaeil are nouns (Éireannach can be an adjective too) meaning 'Irish person(s)', so you can't really use that structure with mo chroí is m'anam. Except perhaps in a figurative sense.

I did’t realise they could refer only to Irish people.

Perhaps something more along the lines of “is as Éirinn iad mo chroí is m’anam”

What about this

Ní Éireannach mé, ach tá croí agus anam an Éireannaigh ionam

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ЯГОН ТОҶИК НЕСТ ИНҶО???

The grammar is correct but the meaning is:
Ní Gael mé, ach tá mo chroí agus m'anam Éireannach
I am not a Gael, but my heart and soul are Irish.

Gael (noun) and Gaelach (adjective) include all Gaelic speaking people, including the people of Scotland and the Isle of Man, not just Ireland.
gaelach with a lower case initial means "homely, pleasant; common, ordinary."
Éireannach (noun and adjective) refer to the culture and people of Éire.

I'd be inclined to align it one way or the other:
Ní Gael mé, ach tá mo chroí agus m'anam Gaelach.
or
Ní Éireannach mé, ach tá mo chroí agus m'anam Éireannach.

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[hr]Múinteoir Gaeilge - Irish Teacher[/hr]
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

We have, in the past, encountered people who weren't comfortable with people who were neither born in nor citizens of Ireland using "Éireannach," which is why we've tended to lean toward "Gael/Gaelach" for people of Irish heritage.

Redwolf

That might be the case if someone were claiming to BE Irish but in this case, the person is specifically stating that they are NOT Irish.

There is a difference between someone saying "I am Irish" and "I am Irish at heart."

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[hr]Múinteoir Gaeilge - Irish Teacher[/hr]
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).