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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?

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

The ST:TNG Technical Manual's description of phaser operation clearly states that the beam will

"travel at c to the target".

Disclaimer: I do not have the book in my possession. Quoting from the Google books version.

In "Wink of an Eye," we see how really slow the beam from a hand-held phaser is (at least, in an atmosphere). When Kirk attempts to "stun" Deela (the "Queen" of the accelerated Scolosians), she is able to slowly step to the side of the beam and allow it to pass by her in order to evade it. Since she can't have been accelerated all that much (at most by a factor of, say, 10,000 times - since at most two subjective days transpire for Kirk in the accelerated state, which would translate to 17 seconds of "ship time" during which Spock, McCoy, and Co. manage to a) realize that the ship has been invaded, and b) whip up a solution), this gives us an upper limit to the beam's speed of propagation.

Since characters in all of the Star Trek series are shown ducking from and dodging phaser beams, it would seem that their speed is somewhat less than the speed of light and often slower than the speed of a bullet from a projectile weapon. If,as the Voyager Technical Manual states phaser are indeed light speed or even relativistic speed weapons missing with them or dodging their beams would be impossible.

Simply put when someone aims and fires a phaser, there is no way for it not to strike its intended target unless the weapon was deliberately aimed away from them. The weapon has no recoil and it would be as if you shined a flashlight at someone (e.g. you can’t miss them)

While phasers travel at the speed of light, that does not mean the person firing it has good aim. If you saw a phaser pointed at you, you'd leap too. Granted, that implies the crews of the Enterprises (and Voyager/DS9) came up against some of the worst shooters in the universe who only manage to kill red shirted ensigns, and true, there have been some laughable 'he dodged a phaser' scenes, but the original series was "Wagon Train" in space so I kind of like when ST uses a bit of old western flair in their story telling.