σπεύδομεν αἵδ᾽ ἀφελεῖν τινὰ τάσδε μερίμνας,
θεῶν δ᾽ ἀτέλειαν ἐμαῖς μελέταις ἐπικραίνειν,
μηδ᾽ εἰς ἄγκρισιν ἐλθεῖν:
Eum. 360-364

Shouldn't ἐμαῖς μελέταις be in the genitive case or "an exemption from my concerns"? I cannot find a usage of the dative that works in this context. It cannot be because ἐπικραίνειν is a compound verb because it is transitive here and has an object in the accusative.

Here is how I read it so far:

"We hasten (not sure how to translate αἵδ᾽ here) to take away these cares from someone and to accomplish from the gods an exemption in ( ? ) my concerns and not to go into question."

What usage of the dative are you taking this to be?

The "in" usage, as the word εν that takes dative. But I put a question mark in brackets because I was not sure if that was a correct translation.

Back in school they used to talk about an "instrumental dative" (δοτικὴ τοῦ ὀργάνου) meaning the means by which something is achieved, in our case the ἀτέλεια of gods.

But doesn't the meaning "in" achieve the instrumental as well?

For example, if one say he speaks in Greek, isn't that basically the same thing as saying he speaks by using or thro the means of or with Greek?