We present a study of viscous overstabilities that can arise in a narrow ring whose particles are tightly packed, so that the pressure tensor exhibits a behavior different than that in a more dilute ring. The main objective of this work is to investigate the role of viscous overstabilities in the excitation of (azimuthal) eccentric modes in narrow rings. The problem is studied through an analytical study of a 2-streamline model and supported by a 10-streamline numerical simulation. There are two possible regimes of instabilities, one in which the mean eccentricity of the ring (i.e., of its azimuthal mode) decreases to a small but finite and nearly constant value, while internal modes of libration reach comparable amplitudes ("small eccentricity regime"), and the other one in which the mean eccentricity of the ring increases to a much larger asymptotic value ("large eccentricity regime") while internal librations are strongly reduced, but not fully damped. This is to be contrasted with the behavior of viscously stable rings, in which both the librations and the mean eccentricity are fully damped, if one neglects the role of the shepherd satellites. Whether one or the other of these regimes is obtained depends on the initial conditions, but the final state in each regime is controlled by the viscous and self-gravitational evolution and not by these initial conditions. In both regimes, there is no rotating frame in which the ring looks stationary, in opposition to the assumption made in data analyses and in previous theoretical modeling. The large eccentricity regime is generically able to produce rings with stabilized mean eccentricities and mean eccentricity gradients quite similar to those of the eccentric rings of Uranus; however, this regime cannot be reached from an initially circular state, so that if viscous overstabilities alone are to account for the eccentricity of narrow rings, then the ring material must have been in eccentric motion at the time of formation, The residual librations in the large eccentricity regime may explain the problems of the standard self-gravity model for the rigid precession for the ε ring, but this appears much more difficult to accomplish for the α and β rings; these residual librations may account for some of the kinematic residuals of the uranian rings, and may also relate to the absence of width-longitude relation in the smaller eccentric rings.