Don't separate them thinking that's going to fix it, because the problem doesn't disappear just because they're together or alone. What's happening is that your male is in full protection mode for his mate and territory, and that gets amplified if the cage has enclosed spaces where they might think there's a potential nest (boxes, tunnels, dark corners). Even though others have already mentioned territoriality, there's something that doesn't quite add up: a lot of the time the aggression gets worse because seasonal changes or even the light hours in the cage are simulating breeding conditions.
Here's the trick that actually works: try introducing your hand in a very different way. Instead of opening it like you're cleaning or handling, bring it with your fingers closed (a loose fist) or carry something in your hand, like a toy or food. Birds that are in defensive mode have fewer triggers if your hand doesn't "invade" their space the traditional way. More subtle changes: get rid of closed boxes or nesting spaces in the cage, increase the light hours (especially if we're already in October with shorter days), and avoid handling during the times when they'd normally be most active reproductively (it's usually between afternoon and dusk).
If he keeps biting you without changes after doing this, then it's worth checking if there's something medical (inflammation, parasites, chronic stress), but for now focus on these behavioral and handling variables. With birds it's much more about reading their context than training them directly.