Following the discovery of radio pulsars at the position of Fermi-LAT unassociated sources by the TRAPUM group, we conduct Swift-XRT observations of six of those 4FGL sources to determine if any pulsar-like X-ray sources are present and to confirm the reported detection of an X-ray counterpart via eROSITA at 4FGL J1803.1−6708. At two of the six targets, we detect no X-ray sources at the TRAPUM radio position, placing an upper limit on the 0.3–10.0 keV flux. At 4FGL J1803.1−6708 we find an X-ray source at the TRAPUM and eROSITA position. At 4FGL J1858.3−5424 we find a new X-ray counterpart at the TRAPUM position with signal-to-noise ratio S/N = 4.17, but also detect a distinct and separate X-ray source. At 4FGL J1823.8−3544 and 4FGL J1906.4−1757 we detect no X-ray flux at the TRAPUM positions, but we do detect separate X-ray sources elsewhere in the Fermi error ellipse. At these last two targets, our newly detected Swift sources are possible alternatives to the radio pulsar associations proposed by TRAPUM. Our findings confirm several of the discoveries reported by the TRAPUM group but suggest that further observations and investigations are necessary to confirm the low-energy counterpart of several unassociated sources.