The beginning of Joel, chapter 1 verse 4, from PUL's Cork Irish Bible:

"Tá are you fhág an luschnuimh ite ag an lócast"

Translation:
That which the palmerworm has left, the locust has eaten

Here, its is the verbal adjective of the verb "to eat." Can somebody explain to me why the verbal adjective form is correct here?? In general, I'm struggling with identifying when to use verbal adjective and verbal noun form of verbs... Does English even have these forms? Irish verbs are so confusing!!

You have a few errors in what you have written above. The verse begins: Tá ar fhág an luschnuimh ite ag an lócast.

The verbal adjective is appropriate here as the English translation would use the past participle of the verb "to eat", i.e. "eaten". The structure itself, "[substantive verb] [something] [verbal adjective]", is very common in Irish. In this case "[tá] [ar fhág an luschnuimh] [ite]" is more complex than usual, because the [something] is actually a separate clause introduced by the relative particle, ar. More typical examples would be like "[tá] [sé] [déanta] agam", "I have it done".

Broadly speaking you can expect the verbal adjective in Irish where you would have the past participle in English, and the verbal noun where you would have the present participle in English. i.e. ag ithe = "eating", but ite ag "eaten by".

Yes, English has both verbal nouns and verbal adjectives, although they are significantly less common in English than in Irish, as they do not fulfil the same function in English. Because they are used in place of present and past participles in Irish, every verb needs both a verbal noun and verbal adjective, whereas in English, they are coined only where they serve some other function. For example, "asking" is generally a present participle of the verb "to ask", however, it can be use as a gerund (i.e. a verbal noun) in phrases like "do you mind my asking". Similarly, any adjective which is based on a verb, is a verbal adjective, hence, the adjectives in examples like "a broken promise" and "a running tap" are verbal.

Wow, you're awesome. It's going to take a lot of time for me to understand your whole post but it was definitely helpful.

If you don't mind, I have another question:

Verbal nouns also confuse me. Consider the following sentence
"D'iarras orthu siúcra agus plúr do cheannach dom"

Here was use the verbal noun "ceannach". Why is this a verbal noun? it looks like it's functioning like a normal verb to me... I don't see anything that makes the word ceannach look like it's being used as a "person, place or thing", which is my understanding of what a noun is.

D’iarras orthu siúcra agus plúr do cheannach dom. = I asked them to buy me sugar and flour.
D’iarras orthu = I asked them
siúcra agus plúr do cheannach dom = to buy me sugar and flour. (lit.: sugar and flour to buy to me)

There is "to buy" in English, an infinitive. In such cases, there is the verbal noun in Irish, hence ceannach.
Word order is different: to buy sugar = siúcra a (< do) cheannach ("sugar to buy")

Perhaps it may be more intuitive for you if you understand the function of do cheannach. In Standardised Irish, and other dialects than Munster, this would be a cheannach lit. "its buying", i.e. "the buying of it". Hopefully this clarifies how it is, in fact, acting as a noun in Irish syntax.

No, it is not "its buying"
"A" between the nominal object of verbal noun and the verbal noun is just a weakened form of "do" (to).
siúcra a cheannach < siúcra do cheannach

But, of course, both have the same form "a".

You're right. I'm conflating possessive a (as in bhí orm a chur...) with prepositional a (as in síol a chur).

That is admittedly harder to convey in English in an intuitive manner. It would be something like uisce a ól lit. "water of drinking" or siúcra a cheannach lit. "sugar of buying".