A ZTF Search for Circumstellar Debris Transits in White Dwarfs: Six New Candidates, one with Gas Disk Emission, identified in a Novel Metric Space

White dwarfs (WDs) showing transits from orbiting planetary debris provide significant insights into the structure and dynamics of debris disks. This is a rare class of objects with only eight published systems. In this work, we perform a systematic search for such systems within 500 pc in the Gaia-eDR3 catalog of WDs using the light curves from the Zwicky Transient Facility (ZTF) and present six new candidates. Our selection process targets the top 1% most photometrically variable sources identified using a combined variability metric from ZTF and Gaia eDR3 photometry, boosted by a metric space we define using von Neumann statistics and Pearson-Skew as a novel discovery tool to identify these systems. This is followed by optical spectroscopic observations of visually selected variables to confirm metal pollution. Four of the six systems show long-timescale photometric variability spanning several months to years, resulting either from long-term evolution of transit activity or dust and debris clouds at wide orbits. Among them, WD J1013-0427 shows an indication of reddening during the long-duration dip. Interpreting this as dust extinction makes it the first system to indicate an abundance of small dust grains (radius $\lesssim$$0.3~{\rm μm}$) in the occulting material. The same object also shows metal emission lines that map an optically thick eccentric gas disk orbiting within the star's Roche limit. For each candidate, we infer the abundances of the photospheric metals and estimate accretion rates. We show that transiting debris systems tend to have higher inferred accretion rates compared to the general population of metal-polluted WDs. Growing the number of these systems will further illuminate such comparative properties in the near future. Separately, we also serendipitously discovered an AM CVn showing a very long-duration outburst $-$ only the fourth such system to be known.