The overall synoptic set-up shows us a strong STJ s/w across the NW Gulf of Mexico at 36hrs:
36hr ETA valid 00z SUN 2/1:

By 54hrs (18z SUN), the STJ s/w begins to weaken in response to the building ridge across the eastern half of the united states, perpetuated in part by the additional energy digging into the central plains and departing 50/50 low.

The surface low pressure area remains rather weak and moves northeastward from the central gulf beginning 0z Sunday and crosses the panhandle by the early evening hours on Sunday.
36hr ETA (SLP/6hr precip) valid 0z Sunday.

54hr ETA (SLP/6hr precip) valid 18z Sunday.
[/img]http://www.nco.ncep.noaa.gov/pmb/nwprod/analysis/namer/eta/12/images/eta_p06_054s.gif[/img]
Notice that 850mb temps surge to well over +10C by 18z Sunday evening across all of central and southern Florida.

ML CAPE values of between 500-1000 J/kg (surface-based convective available potential energy measured in Joules per kilogram) on average combined with good storm-relative helicity values of over 250 m^2/s^2 across central and southern Florida should support a few strong/rotating convective updrafts in the warm sector.
http://weather.unisys.com/eta/misc/eta_cape_4panel.html
http://weather.unisys.com/eta/misc/eta_hel_4panel.gif
Although with that said, the ETA implies that the best SRH does not geographically coincide with the strongest surface based and mixed layer CAPE. So with this in mind, I’m more inclined to think that pulse variety thunderstorms are more likely, with a few tornado thunderstorms possible as well but a widespread/major outbreak is NOT likely IMO.
UVM should be sufficient to compensate for the slightly weaker CAPE values further north along the warm frontal boundary to lift parcels to where they need to be in order to initiate convection. Convective temps will also be met. This could be the area which sees the best tornadic potential given the VERY strong SRH values approaching 1050 m^2/s^2 across the region and the strong lift ahead of the s/w.
EHI values (or energy-helicity index values) are generally near 1.00 across most of the central part of the state, which is considered to be the minimum value needed to get rotation.
Areas which see considerable amounts of sun (surface warming) will be most susceptible given the strong influx of moisture and warm advection to see the best action as CAPE values will elevate quickly. Those which do not will be reliant on strong UVV in order to initiate convection.