I am apparently fighting an enemy with hundreds of ships (well, it sure seems that way). I can stay out of their range while still hitting them, with lots of shuffling between waypoints and slowly whittle them down. But I am having to pick new targets for my ships every few turns and thus far, I am doing it manually. Needless to say this is 'extremely' tedious. Is there any button/option I can pick to have my ships switch automatically to a new target once their existing target has been destroyed?
No. In the next version there will be though.
You are using the "assign fleet BFC" button though right? That at least means you can retarget the entire fleet in one click - just click it when any ship in the fleet is selected whenever an enemy ship has been destroyed and all ships without a target will find new ones