There's good reason for concern. As Dave Robinson reports in his May 4, 2013 summary,
...low temperatures were at or below freezing at one or more locations on 16 April days. Six days saw a location fall to 25° or colder. The first of these days was the 2nd, when cold air drainage resulted in the valley locations of Walpack (Sussex) dropping to 18° and Pequest (Warren) 20°. On the 3rd, Pequest reached 21° and Kingwood (Hunterdon) 22°.Radiation frosts happen when freezing conditions occur on clear nights with little or no wind, when the outgoing radiation is greater than the incoming, and cooling air temperature near the surface creates a stable temperature inversion near the ground. These same conditions create a crystal clear atmosphere, which makes a full moon appear exceptionally bright.
But, is there evidence that a frost is more likely to occur on a night with a full moon than one without?