The detection of GeV {gamma}-ray emission from Galactic novae by the Fermi-Large Area Telescope has become routine since 2010, and is generally associated with shocks internal to the nova ejecta. These shocks are also expected to heat plasma to ~10^7^K, resulting in detectable X-ray emission. In this paper, we investigate 13 {gamma}-ray emitting novae observed with the Neil Gehrels Swift Observatory, searching for 1-10keV X-ray emission concurrent with {gamma}-ray detections. We also analyze {gamma}-ray observations of novae V407 Lup (2016) and V357 Mus (2018). We find that most novae do eventually show X-ray evidence of hot shocked plasma, but not until the {gamma}-rays have faded below detectability. We suggest that the delayed rise of the X-ray emission is due to large absorbing columns and/or X-ray suppression by corrugated shock fronts. The only nova in our sample with a concurrent X-ray/{gamma}-ray detection is also the only embedded nova (V407 Cyg). This exception supports a scenario where novae with giant companions produce shocks with external circumbinary material and are characterized by lower density environments, in comparison with novae with dwarf companions where shocks occur internal to the dense ejecta.