As main-sequence stars with C$>$O, dwarf carbon (dC) stars are never born alone but inherit carbon-enriched material from a former asymptotic giant branch (AGB) companion. In contrast to M dwarfs in post-mass transfer binaries, C$_2$ and/or CN molecular bands allow dCs to be identified with modest-resolution optical spectroscopy, even after the AGB remnant has cooled beyond detectability. Accretion of substantial material from the AGB stars should spin up the dCs, potentially causing a rejuvenation of activity detectable in X-rays. Indeed, a few dozen dCs have recently been found to have photometric variability with periods under a day. However, most of those are likely post-common-envelope binaries (PCEBs), spin-orbit locked by tidal forces, rather than solely spun-up by accretion. Here, we study the X-ray properties of a sample of the five nearest known dCs with $Chandra$. Two are detected in X-rays, the only two for which we also detected short-period photometric variability. We suggest that the coronal activity detected so far in dCs is attributable to rapid rotation due to tidal locking in short binary orbits after a common-envelope phase, late in the thermally pulsing (TP) phase of the former C-AGB primary (TP-AGB). As the radius of a TP-AGB star rapidly expands once it reaches the C giant phase, the initial range of orbital periods that can lead to the mass transfer balance necessary to form a short period dC remains a mystery.