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Priority Sector deals %HP damage which is quite significant. It also occurs to every enemy squadron that is in range of the sector'd side, so you can deal a lot of damage when it affects more than one air group.
In general, the %HP damage makes it always worth using.
Buffing one side does decrease the other side, but as long as you are shooting at least 1 squad it'll work out.
Exmaple: Ship has 100 AA dps and using Priority Sector on the left side. There are planes Left and Right of the ship.
150% left side = 150 dps
50% right side = 50 dps
If there was a squadron on the left and the right, you're still dealing 200 dps overall which would be the same as if you didn't use Priority Sector.
Not at all. The AA strength just returns to normal (100% on both sides) during the cooldown period. You can verify this by hitting "Alt" in client and it'll show you.