A phaser fires a nadion particle beam at its target. That sounds like inertial mass is involved so it doesn't reach light speed. Semi-out-of-universe we see how fast a phaser beam is: I'd estimate something around 80 kph. You can see it moving with the naked eye. But that might be just a way to tell a story; to establish causality to the audience: "Look! First the beam is here, now it's here, now something's exploding." So the audience knows why the enemy ship explodes: It was a phaser beam, going from here to there. So maybe we are just shown a "symbolic slo-mo" to let us know what's happening.
Are there any in-universe insights on how fast a phaser beam travels? E.g. can I escape (escape, not evade) one on full impulse?
Answer
The "Voyager Technical Manual" (written by longtime Star Trek production staffers Michael Okuda and Rick Sternbach as an 'aide-memoire' for potential scriptwriters) specifically states that phaser beams travel at the speed of light.
This is backed up by another quote from the earlier "TNG Technical Manual" which clearly states that the beam travels at "c"
Given that full impulse is supposedly well short of the speed of light, the short answer to your question is "no, you cannot escape a phaser beam by travelling at sublight speeds".
Travelling at warp speed would be considerably more effective against a foe travelling below lightspeed since you would simply outrun the beam (unless the phaser is fired from a ship travelling at a similar warp factor).
Out-of-universe (e.g. from a TV production standpoint) phaser beams do often travel considerably slower.
As the video below shows, beam speed is wildly inconsistent. At the slowest they seem do seem to be moving about a couple of feet per frame. At the fastest, the beam is instantaneous from one shot to the next
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