The answer to this question is negative, if "almost tidally locked" signifies a small but persistent deviation from an exact resonance. The anwser, however, is positive if "almost tidally locked" implies large-magnitude libration.
Within a tidal two-body problem, a secondary body experiences two torques exerted on it by the primary: a tidal torque ${\cal{T}}_{tide}$ and a torque generated by the permanent dynamical triaxiality of the secondary's figure, ${\cal{T}}_{tri}$ (in practitioners' argo, the "triaxial torque").
For bodies that are rigid or have a solid crust, the triaxiality-caused elevation above the mean level is typically much larger than the hight of the tidal bulge. As a result of this, in a spin-orbit resonance (a synchronous resonance, like the Moon; or higher resonance, like Mercury), ${\cal{T}}_{tri} \gg {\cal{T}}_{tide}$.
Two important caveats are in order, though:
(a) In a spin-orbit resonance, the rotator is subject to forced physical libration of a magnitude $\simeq e^2$.
(b) Outside spin-orbit resonances, ${\cal{T}}_{tri}$ averages out due to rotation, and $ {\cal{T}}_{tide}$ plays a key role.
Usually (the Moon being a notable exception), forced physical libration is predominantly longitudinal. Its evolution is analogous to ordinary pendulum. The torque ${\cal{T}}_{tri}$ splits into two parts, one playing the role of restoring force, another of exterior forcing. Like a pendulum, the rotator is impelled to deviate from the exact equilibrium position, but is pushed back by the restoring part of the torque.
For gas and liquid bodies lacking dynamical triaxiality, we have ${\cal{T}}_{tri} =0$. So dynamics is governed by ${\cal{T}}_{tide}$. Calculation demonstrates that in this situation the rotator always ends up in a so-called pseudosynchronous state, a regime wherein the rotator has an excess angular velocity (as compared to the mean motion $n$) proportional to $6ne^2$. This calculation is presented, with an error, in this standard text, eqn (5.14). A correct calculation can be found in our paper.
(Be mindful that the value of said excess in angular velocity is rheology-dependent.)
(2) Now, do pseudosynchronous rotators in the solar system exist? Not to the best of my knowledge.
(3) Are there intensely librating objects? Yes, there are.
Owing to its potatoe shape, Phobos is librating in longitude with a magnitude of $1.2^{\,o}$. Epimetheus has a more regular shape but a larger dynamical triaxiality due to its internal inhomogeneity, so its longitudinal libration magnitude is $5.9^{\,o}$. This is a lot. Libration more than doubles tidal dissipation in Phobos, and boosts tidal dissipation in Epimetheus by a factor of 26 -- more than an order of magnitude.
