I think there is now plenty of evidence for the evolving rotation period of the Earth and evidence for a tidal torque increasing the Earth-Moon distance.
But was the Moon always in synchronous orbit with the Earth or is there any evidence that its rotation period was different to its orbital period in the (distant) past?
There is plenty of evidence for the Moon's orbital period being different in the distant past. However, there is little if any evidence that the Moon's rotation rate differed from its orbital rate.
The Moon is hypothesized to have formed very close to the Earth (four to six Earth radii) when the Earth was still very young -- and very hot. It is also hypothesized that the Moon became tidally locked extremely soon after it formed, perhaps as little as 100 days. The heat of formation and the close proximity to a very hot Earth would have made the ratio of the Moon's quality factor $Q$ and the Moon's second degree fluid Love number $k_2$ (i.e., $\frac{Q}{k_2}$) very small. This ratio plays a strong role in the tidal locking time. The close proximity also plays an extremely strong role in the tidal locking time.
If the Moon was indeed tidally locked in as little as 100 days, and if the Moon was still covered by a magma ocean at that time (as is widely hypothesized), there is no hope for evidence that the Moon had a different rotation rate at the time of its formation. However, the odds of the Moon having formed tidally locked from the very onset are near nil. One hundred days is just past the very onset, but it is not the very onset. That mere 100 day locking time still provides more than enough time for the Moon to have developed very distinct near sides and far sides.
There is indirect evidence of tidal locking elsewhere in the solar system. Ganymede, Europa, and Io are tidally locked to Jupiter. These three moons are more or less in a 1:2:4 orbital resonance. This, coupled with tidal torques, result in an interesting hysteresis for Io. The tidal forces and torques act to circularize Io's orbit, while the orbital resonances act to increase Io's eccentricity. Io heats up internally when its orbit becomes more eccentric. This causes the extreme vulcanism seen on Io. A body in an eccentric orbit cannot have perfectly synchronized rotation rates and orbital rates as it takes significantly more time to synchronize than it does to go from periapsis to apoapsis (and back). This increased eccentricity eventually enables tidal forces and torques to dominate over orbital resonance forces and torques. Io's orbit eventually becomes more circular, resulting in Io cooling off, which in turn causes $\frac{Q}{k_2}$ to increase. This in turn results in the resonance forces and torques once again dominating, for a while. It's a rather interesting hysteresis loop.
No, there is no evidence of a period when the Moon wasn't tidally locked. However, I don't think it is possible to answer such a question with an unequivocal 'no', since it is hard to rule out the existence of evidence for a given hypotheses. It is like trying to prove a negative.
If the Moon tidally locked as it was formed, then it has almost certainly always been tidally locked. There are hypotheses of such formation in accepted scientific literature. From Roy et. al EARTHSHINE ON A YOUNG MOON: EXPLAINING THE LUNAR FARSIDE HIGHLANDS:
Our hypothesis requires that the Moon formed effectively tidally locked.
If the Moon didn't form tidally locked, as it locked its orbit would have shrunk or grown (depending on if its initial rotation was prograde or retrograde). Unfortunately, this would have happened so soon after formation that any geological evidence has long been swallowed via the mechanism of tectonic plates. The oldest Earth surface rock is less than 3 billion years old. As for some evidence for a period of non-tidal lock on the Moon itself, I can't think of any.
