We present results from a new pipeline custom-designed to search for faint, undiscovered solar system bodies using full-frame image data from the NASA Transiting Exoplanet Survey Satellite (TESS) mission. This pipeline removes the baseline flux of each pixel before aligning and co-adding frames along plausible orbital paths of interest. We first demonstrate the performance of the pipeline by recovering the signals of three trans-Neptunian objects -- 90377 Sedna ($V=20.64$), 2015 BP519 ($V=21.81$), and 2007 TG422 ($V=22.32$) -- both through shift-stacking along their known sky-projected paths and through a blind recovery. We then apply this blind search procedure in a proof-of-concept survey of TESS Sectors 18 and 19, which extend through a portion of the galactic plane in the Northern Hemisphere. We search for dim objects at geocentric distances $d=70-800$ au in a targeted search for Planet Nine and any previously unknown detached Kuiper belt objects that may shed light on the Planet Nine hypothesis. With no input orbital information, our present pipeline can reliably recover the signals of distant solar system bodies in the galactic plane with $V<21$ and current distances $dlesssim 150$ au, and we elaborate on paths forward to push these limits in future optimizations. The methods described in this paper will serve as a foundation for an all-sky shift-stacking survey of the distant solar system with TESS.
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