We present follow-up observations of the serendipitously discovered 7-s X-ray pulsar AX J1845-0258, which displays characteristics similar to those observed in the anomalous X-ray pulsars (AXPs). We find a dramatic reduction in its 3-10 keV flux in both new ASCA and RXTE datasets. Within the pulsar's position-error locus, we detect a faint point source, AX J184453-025640, surrounded by an arc of diffuse X-ray emission. This arc is coincident with the South-East quadrant of the radio shell of the newly discovered supernova remnant G29.6+0.1, reported in our companion paper (Gaensler et al. 1999). Lack of sufficient flux from the source prevents us from confirming the 7-s pulsed emission observed in the bright state; hence, at present we cannot definitively resolve whether AX J1845-0258 and AX J184453-025640 are one and the same. If they are the same, then the peak-to-peak luminosity changes recorded for AX J1845-0258 may be larger than seen in other AXPs; closer monitoring of this pulsar might lead to a resolution on the mechanism that drives AXPs. ; Comment: 5 pages with 2 figures, LaTex, emulateapj.sty. To appear in the Astrophysical Journal Letters