Our survey of the fourth Fermi Large Area Telescope catalog (4FGL) unassociated gamma-ray source regions using the X-Ray Telescope (XRT) and Ultraviolet/Optical Telescope (UVOT) on board the Neil Gehrels Swift Observatory (Swift) provides new XRT and UVOT source detections and localizations to help identify potential low-energy counterparts to unassociated Fermi gamma-ray sources. We present a catalog of 218 singlet and 70 multiplet Swift X-ray sources detected within the positional uncertainty ellipses of 244 unassociated Fermi gamma-ray sources from the 4FGL-DR4 catalog, 144 of which are not previously cataloged by S. Kerby et al (2021ApJ...923...75K, Cat. J/ApJ/923/75). For each X-ray source, we derive its X-ray flux and photon index, then use simultaneous UVOT observations with optical survey data to estimate its V-band magnitude. We use these parameters as inputs for a multilayer perceptron neural network classifier (NNC) trained to classify sources as blazars, pulsars, or ambiguous gamma-ray sources. For the 213 singlet sources with X-ray and optical data, we classify 173 as likely blazars (P_bzr_>0.99) and six as likely pulsars (P_bzr_<0.01), with 34 sources yielding ambiguous results. Including 70 multiplet X-ray sources, we increase the number of P_bzr_>0.99 sources to 227 and P_bzr_<0.01 sources to 16. For the subset of these classifications that have been previously studied, a large majority agree with prior classifications, supporting the validity of using this NNC to classify the unknown and newly detected gamma-ray sources.