Does Agony's Embrace exhaust after use?

No — Agony's Embrace never exhausts, because it's a Legend with a triggered ability, not a permanent with an activated ability.

It sits in the Legend Zone (it can't be moved, killed, or exhausted at all — Rule 174.3), and its text is "Once each turn, when you play a card from face down, draw 1. Then, if it's not your turn, discard 1." There's no [T] exhaust symbol and no cost to pay, so "using" it just means the trigger fires automatically the first time each turn you play a card from face down — no exhaust, no once-per-turn tap limitation.

The "once each turn" is the only usage limiter, so it can trigger on your turn and again on an opponent's turn (that's what the "if it's not your turn, discard 1" clause is compensating for).