Outer solar system satellites are subjected to strong tides that guide their evolution. Specifically, how a moon evolves depends sensitively on feedbacks between its orbit, the thermal state of its interior, and the extent of dissipation in the moon, which can be affected by resonances with other moons as well as the primary planet. Impacts, while an ubiquitous planetary process, are generally neglected in models of tidal evolution because impact heating from an individual event should be removed relatively quickly and is spatially limited such that it is unlikely to affect global dissipation. However, studies have investigated the effects of a period of enhanced bombardment on differentiation of an ice-rock moon (Barr and Canup, 2010), vaporization and destruction of an icy moon (Nimmo and Korycansky, 2012), and enhanced tidal heating due to an impact-induced increase in eccentricity (Zhang and Nimmo, 2012). To date, no studies have examined how tidal dissipation in an ice shell would change after being warmed by a period of enhanced bombardment. Here, we track the thermal evolution of an ice shell, as well as the eccentricity evolution of the orbit, for two generic icy moons (one large and one small) under the influence of global, heavy bombardment. As a proxy for impact heating, which we do not directly model, we introduce a layer of warm ice, with characteristics informed by the results of impact modeling into ice shells (e.g., Roberts and Stickle, 2021), and vary both its depth and radial extent. Our results suggest that, under certain conditions, impact heating does have substantial, long-term effects on the thermal-orbital evolution of an icy moon. We discuss observational constraints, in the form of heat flow measurements and interpretations of crater relaxation on icy moons, that could be diagnostic of past epochs of enhanced heat flow due to impact-generated warming of an ice shell.