NEVER ignore hits on your shields thinking, 'I'll just kill this guy first' ..
Disengage and deal with the attack imo.
Key issue, right there.
Maybe a bit "immersion breaking" but it helps to understand how the game works.
When a CZ kicks off, the AI's all "pair off" into their own little fights.
Wait until the fight starts before picking a side and all the AIs will already have picked their "partner".
Pick a ship to attack and join in to help destroy it.
Once you've done that, a new enemy is likely to spawn and, when they do, they'll be looking for a "partner" to fight against.
If you get attacked, it means an enemy ship
HAS picked you to be their "partner" so they're going to follow you around, attacking you.
That's ship should be your immediate priority.
Kill it and you can go back to "ganging up" on an enemy who's already fighting somebody else.
It's also the reason why you should try to avoid getting right into the middle of a fight, especially if your side is losing.
If there's a whole bunch of enemies who kill their "partner", they'll all be looking for a new one and several might all decide to pick you.
As a rule, I find the simplest way to have an easy life in CZs is to just stick around other green blobs.
Gives you strength of numbers.
Minimises the chances of
you being singled-out for attack.
Allows you to shoot at anything friendly ships are shooting at.