The tech manuals can be helpful in filling in blanks, but they have been contradicted on numerous occasions by the episodes themselves. And if we're going to include the tech manuals for TNG, then what about the writer's bible for TOS? Phasers were originally described as an offshoot of the asteroid deflector beam technology which a starship much apply when at warp. As the deflectors must by neccessity be FTL, so must the phasers.
Secondly, starfleet officers rarely target their foes manually (indeed, when Sulu first does this in TWOK the results are less than stellar!). Instead, a character will lock phasers
on target - in other words, the computers and sensors do the job - and they should have no trouble tracking a FTL opponent, since they themselves are a FTL body. All the phaser turrets need to do is target an area ahead of the destination, controlled precisely by the same systems.
A final word on the seemingly massive ranges employed by TOS - at the time of the series production, typical aircraft carriers and gunboats ranged their weapons over many miles, some even to the limit of the horizon. Would it not be natural to extrapolate these distances into their outer-space equivalent?