Learn Arabic (Modern Standard): فاض قلب مريم شوقا.

This is the Accusative of Specification (التمييز). After verbs expressing filling, increasing, or overflowing like فاض (to overflow), Arabic does not usually use a preposition. Instead, the noun explaining exactly what is overflowing is put in the accusative case, which is why شوقا (longing) takes the -an ending.

Why is the verb فاض masculine when the sentence is about Maryam?

The subject of the verb is قلب (heart), which is a masculine noun in Arabic. The verb always agrees with the grammatical gender of the subject itself, not the person who owns it.

Since Maryam owns the heart, shouldn't مريم be in the genitive case with a kasra (-i) ending?

It is in the genitive case because it is the second part of an Idafa (possession) structure! However, female names like مريم are diptotes, meaning they are restricted from taking standard endings. In the genitive case, a diptote takes a fat-ha (-a) instead of a kasra (-i).

How do I know the root of the hollow verb فاض so I can look it up in a dictionary?

فاض is a hollow verb with an alif in the middle. To find its true root, you can look at its present tense form يفيض (it overflows) or its verbal noun فيضان (flood). This reveals that the hidden middle root letter is a yaa, so you would look it up under faa-yaa-daad.