The steering direction will be dependent on the relative juxtaposition of the greatest diabatic heating within the overall low pressure gyre plus synoptic-scale flow pattern. (For example if the greatest heating was further NW towards the Carolina coast but the synoptic-scale flow favored a SSW-SW to NNE-NE propagation, the net contribution of these would be likely somewhere in between (north-ish), however would be dependent on the magnitudes of the diabatic heating produced by the convection and speed of the mid-upper level winds. The biggest implication of losing the spurious vorticity lobes off FL and seeing a shift towards a more consolidated vorticity max over the Central Gulf coast will aid in moisture transport further west into the Carolinas and could shift the primary low in the Atlantic closer to the climatologically preferred area of highest baroclinicity near the interface of the Gulf stream and adjacent cooler continental shelf waters. Akin to what we often observe in overrunning events, large, intense clusters of convection along the Gulf Coast block the northward transport of moisture into the SE US by altering local winds in the low-levels and inducing large-scale sinking in the immediate vicinity, and if we get too much convection well offshore earlier on it hurts our chances for snow by blocking low level moisture (which is why there's such a stout dry layer on the GFS). However, on the other hand, in the cases of Miller A cyclones, large clusters of Cb convection (as observed in January 2000) can aid in moisture transport in the long-run if properly placed in the flow regime (in the case of Jan 2000 it was further NW towards the NE Gulf coast), by leading to larger-scale pressure falls and generation of potential vorticity which foster synoptic-scale areas of low pressure, which, as they intensify, their wind field grows larger and more intense, thus they're capable of transporting more moisture further away from their epicenter. It's definitely a double-edged sword and situationally dependent, but the initial surge of moisture into eastern NC will probably be facilitated by this trend towards a weak Gulf low if it continues...