A straight answer
I'll try to answer this straight, without getting bogged down in side issues that subvert the question, like the difficulty of following the precognition, failures of attention, different thinking speeds, etc.
So, let's assume the precognition is perfect, and followed perfectly and equally well by both. What does this mean? You mentioned chess engines, so let's suppose both combatants have a version of "perfect play" like an ideal chess engine could have.
For this analysis, assume the fight consists of a series of turns - each lasting something like 10 milliseconds - and in each turn each combatant may choose an action - each action being to activate certain muscles. At the end of the fight there is an outcome; each combatant takes a certain amount of injury, perhaps lethal perhaps not. The combatants each have a preference across all possible outcomes; each combatant ranks each outcome from best to worst.
For the precognitive combatants to have "perfect play" can be defined as them choosing the maximin move at each turn. This means they always choose the move that guarantees them the best outcome they can be sure to achieve from the current position, in the worst case of what their opponent might do to thwart them.
Given these premises, the outcome depends on what the weapons are. You said we are talking melee bladed weapons. Any armor? Suppose no armor.
MAD
Suppose the weapon is a knife. Knife fights in real life are often a matter of mutually assured destruction (MAD); either combatant, if they have enough disregard for their own life, could rush in and inflict a serious, perhaps lethal wound on their opponent. It takes time for the reckless aggressor to die from any wound, and in that time they can stab several times.
This is true also for longer weapons. In real-world competitive fencing, fights are determined by who strikes the first blow. But the opponent's strike often comes mere milliseconds after that. If they didn't care about who was first and just charged, they would both hit.
This may not hold up in a precognitive combat (see "mirroring" below). But if it does, then there will never be a one-sided victory where one combatant gets away unscathed and the other is dead. The possible outcomes in a melee MAD scenario are one combatant wins (by virtue of being stronger/faster/taller) but is grievously injured himself; both combatants kill each other; or, knowing that these are the outcomes and neither wanting them, both combatants decline to engage.
There won't be any "testing of defenses." Both combatants know from the start what's going to happen, just like a perfect chess engine knows immediately whether any given position is win/loss/draw with perfect play. All that remains is just to do it.
Large physical disparities
If one combatant has a substantial physical advantage over the other, then that combatant is not going to lose, and probably going to win. They can overpower their opponent, outmaneuver them, or outreach them. Grabbing the opponent's wrist to turn the melee into a wrestling match could be a winning strategy.
Mirroring
A precognitive knife fight is different from a normal one, because it's harder to get past the defender's guard, assuming the defender has something to block with such as a bundled rag or piece of wood. The attacker can't feint one way, then go the other way to get around the defender's guard, because the defender will never fall for a feint. If the defender isn't too different in reach or speed from the attacker, the defender can simply mirror the attacker's motions perfectly, so the defender's blocking hand will always be there to meet the attacker's weapon.
So it's plausible that a precognitive knife fight between physically fairly even combatants who have something to block with, would be a draw, with neither able to inflict an injury - or last until the combatants are exhausted and the one who is physically less exhausted wins.
Of course, if neither of the combatants would gain a decisive victory from such a battle of exhaustion, then it won't even start; they'll just agree not to fight at the beginning.
Compounding small advantages
In chess, when a position is close to even, strong chess engines will spend a long time jockeying for position - trying to compound small positional advantages that might eventually result in a win. (Although: when the position is close to even, it usually just is even, and the outcome is a draw regardless of this jockeying.)
What would this look like in a knife fight? One of the combatants might be a little taller, with longer reach, or they might be a little faster. So they might try to get around their opponent, to make the position asymmetrical. That's the only way they're going to get around the opponent's mirroring defense.
My intuition, picturing this, is that this is usually not going to work. The opponent just needs to take an occasional step back to nullify any such positional jockeying.
This does mean that the combatant with a physical advantage can make his opponent take steps back, and that might be used to corner him, if the terrain is advantageous. On the other hand, if the terrain is open enough, the defender can just retreat in a circle so nothing changes.
Note that making the defender trip over something as he retreats backwards is impossible; they're a perfect precog, so they could fight with their eyes closed and it would make no difference.
Armor
The precog enemy's aim for any gaps in the armor would be perfect. So, in one sense armor may not be as effective in a precog combat as it would be in a normal combat. If you're wearing full plate with a 1 cm eye slit in the helmet, the enemy can just casually stab his sword right into that slit without any hesitation or delay.
A precog combatant doesn't actually need an eye slit, eyes being superfluous compared to his precognitive sense, but the same holds for other small gaps in the armor.
What armor does do is limit the points the enemy can target; only the eye slit, and not the side of the head, for example. This means that the defender only needs to deflect the incoming attack slightly to make it miss; he doesn't have to stop it entirely.
This doesn't mean the defender will spend much time actually deflecting incoming attacks, though. The attacker won't bother making any attack the defender can block.
So, armor makes precog combat even more likely to end in a draw.
Longer weapons
Obviously, a longer weapon is an advantage against a shorter one. If you have a sword and the enemy has a knife, and you're both precogs, you almost certainly win. The enemy can't really mirror your motions because you can whip the point of the sword around faster than he can move his blocking hand, because of the lever arm of the sword.
If both precogs have longer weapons, the battle might be less likely to shed any blood, because parries are easier. A precog knife fight is drawish because both combatants can injure each other, whereas a precog rapier fight is drawish because neither combatant can score a hit.
The precogs aren't likely to waste much time clashing swords and parrying, though. They know their strikes would be parried, so they have little to gain from making them, unless they have more stamina and are trying to exhaust their opponent.
Ranged weapons
You discounted ranged weapons, but they would actually be quite powerful and difficult to dodge. Suppose your precog opponent is 20 ft away shooting an arrow at a respectable 200 fps. You have 0.1 seconds to dodge. If they aimed for your chest, it is not possible to move your chest out of the way in 0.1 seconds. (And remember, the opponent is precog, so unlike with a non-precog opponent, it is not possible to anticipate ahead of time where their aiming point will be and get out of the way before they shoot. They can keep their aiming point dead center on your chest however you move and can't be faked out, so you really do have only 0.1 seconds to react.)
You might try to deflect the arrow instead of dodging it - for example, using your dagger. But the problem there is that the bow-wielding enemy can change his aiming point faster than you can move your hand. He only needs to make a tiny motion to switch his aim from your thigh to your neck, while you have to move your hand a couple feet. So the enemy just needs to flip his aiming point a couple times until your dagger isn't in the way, then shoot.
If you're in full plate armor, however, a tiny motion lasting 0.1 seconds could make his arrow miss your eye slit. Though of course, he's not going to actually shoot in that case; he'll save his shot until he precognitively knows it will hit. If you're armored enough that none of his shots will ever hit, he knows immediately and will just either retreat or switch weapons without bothering to shoot.