We present photometric and spectroscopic studies of the white dwarf (WD) populations in the intermediate-age open clusters NGC 6633 and NGC 7063 as part of the ongoing Lick-Arizona White Dwarf Survey. Using wide-field CCD imaging, we locate 41 candidate WDs in the two cluster fields: 32 in NGC 6633, and 9 in NGC 7063. Spectroscopic observations confirm 13 of these candidates to be bona fide WDs. We describe in detail our Balmer-line fitting technique for deriving effective temperatures and surface gravities from optical DA WD spectra and apply the technique to the 11 DA WDs in the sample. Of these, only two DA WDs are at the cluster distance moduli, one in each cluster. Two more DAs lie 0.75 mag foreground to NGC 6633, raising the possibility that these are double degenerate systems in the cluster. If they are nearly equal-mass binaries, both of these systems likely have combined masses above the Chandrasekhar limit. One DB WD is found to be consistent with membership in NGC 6633, which would make this the third confirmed He-atmosphere WD in an open cluster, although further data are needed to confirm cluster membership. The WD consistent with membership in the cluster NGC 7063 has a low mass (≈0.4 M), suggesting it may be a He-core WD resulting from close binary evolution. Three of the 11 hydrogen-atmosphere WDs in this study are observed to have Ca II absorption; the number of DAZs in this study is consistent with previous observations that ~25% of field WDs are DAZs.