Diarmuid ó Sé's Gaeilge Chorca Dhuibhne has both the forms insíonn (§518) and niseann (§544) and both verbal nouns insint (§571) and nisint (§544) are given. He states that niseann is the form used in the future tense and conditional mood (§518). Does this extend to the use of the verbal nouns? e.g.

Chuin an fhírinne a insint...
Ba mhaith liom an fhírinne a nisint duit

I would guess so, given that I can't imagine when else one would use nisint, unless they're interchangeable? Furthermore, native Munster authors seem to overwhelmingly only use insint on Nua Chorpas, is this a reflection of actual spoken usage or just a result of standardisation of texts?

Any grammar corrections are greatly appreciated as well.

I'd think a verbal noun "nisint" is rare.
(or even totally made up just to fill a gap in the paradigm of the verb "nis").
The verbal noun is usually "insint". /i:n´ʃin´t´/. All the examples he gives contain insint (or dh’insint), not nisint.

There certainly is no kind of "future verbal noun".

The verb form used in the future tense and conditional mood is based on neos- (an ó-future of niseann). So, future is neosaidh, conditional neosadh.
There is no future form inseoidh in the dialect (which would be an ó-future of insíonn)
And there is no future form ineosaidh in the dialect (which would be the traditional Classical Irish ó-future of inis, inseann).

Diarmuid Ó Sé's inclusion of nisint there is itself proof that this exists. His book is based on a number of older speakers mainly born 1910-1930, but the list is at the beginning of the book. It doesn't mean people born in the 1990s would say nisint.

It has to be born in mind that vowels run together in Irish, so it wouldn't be "fhírinne a nisint", but "fhírinne nisint".

The use of insint or nisint doesn't depend on if it is being used in an indicative or conditional sense. So chuin na fírinne a nisint (with fírinne in the genitive) and ba mhaith liom an fhírinne a (dh')insint duit would both work.

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I'm an intermediate speaker of the Corca Dhuibhne dialect of Irish and also have knowledge on the old spelling
Soir gaċ síar, fé ḋeireaḋ thíar

There seems to be both chun an fhírinne a insint and chun na fírinne a insint on Nua Chorpas but more of the former from native Munster authors. Again, I'm not sure if this is due to standardsation, or some exception or something else. Any other input would be greatly appreciated.

Yes, fírinne can be masculine or feminine. I just assumed that you were treating it as masculine as you provided ba mhaith liom an fhírinne a nisint duit. (Though, of course, chuin na fírinne/an fhírinne a insint is considered Béarlachas when compared to the more traditional lé fírinne)

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I'm an intermediate speaker of the Corca Dhuibhne dialect of Irish and also have knowledge on the old spelling
Soir gaċ síar, fé ḋeireaḋ thíar

In An t-Oileánach: chun na fírinne a rádh

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I'm an intermediate speaker of the Corca Dhuibhne dialect of Irish and also have knowledge on the old spelling
Soir gaċ síar, fé ḋeireaḋ thíar