Short answer: As you noted in your question, it depends on what is overwhelming you, and neither interpretation is "wrong." Is it the depth (degree) of emotion, or the emotions themselves?
The least controversial analysis is to strip the modifier. That is, don't let the prepositional phrase (of love, gratitude, and individuality) fool you. The subject-verb agreement depends on the base noun. So you have:
...I could not possibly grasp the depth...that currently overwhelms me.
The base noun is singular. The complex adjectival prepositional phrase that completes the noun phrase is of no import.
The above is true if you mean:
I could not possibly grasp the depth (of love, gratitude, and individuality) (that currently overwhelms me).
But it would be valid to suppose the semantics are:
I could not possibly grasp the depth (of (love, gratitude, and individuality) (that currently overwhelm me)).
In other words, if the phrase that currently overwhelm me is meant to refer to the trio of attributes, then that whole phrase becomes part of the prepositional phrase (beginning with of), and hence the verb should be plural.
This is at least as valid an interpretation as the first option, and frankly sounds more natural to my (AmE) ear.
For reference, see point 5 at this OWL link.