To expound on DavePhD's nice answer, molar conductivity follows Kolrausch's law, $\Lambda_m=\Lambda_m^{\circ} - \kappa \sqrt{c}$ at low concentrations, which means that extensive conductivity follows $\Lambda_m c = c\Lambda_m^{\circ} - \kappa c \sqrt{c}$ in this domain.
As you've discovered this relationship breaks down at high concentration. In a loose analogy to the virial equation of state for gases, I'd propose fitting your data to a modification of Kolrausch's law, specifically, to an expansion in $\sqrt{c}$:
$\Lambda_m=\Lambda_m^{\circ} - \kappa_1 \sqrt{c} - \kappa_2 (\sqrt{c})^2 - \kappa_3 (\sqrt{c})^3 + ...$
I'm not enough of Debye-Huckel theorist to know if this expansion has the same theoretical justification/derivation in terms of intermolecular forces that the virial equation for gases does. But, (a) it definitely introduces extra degrees of freedom that you will need to fit your more complex data, and (b) if your data did follow Kolrausch's law (and the data for e.g. sodium chloride doesn't look too far off, at least by eye), your fit would tell you so, meaning you wouldn't have to convert or translate your fit parameters to a new system of units to compare with tabulated values for Kolrausch's law -- your fit would tell you the Kolrausch parameters.
UPDATE: I dug around in the literature a bit. Wikipedia pointed me towards this 1960s JACS paper. There they give the Fuoss-Onsager equation, which can be written as:
$$\Lambda_m=\Lambda_m^{\circ} - \beta_1 \sqrt{c} - \beta_2 c - \beta_3 c \ln{c}$$
The $\beta_i$ parameters are expanded in the paper to show dependence on some underlying solvent and solute properties, but that isn't important for fitting compound-specific curves to your conductivity data. You can find papers on the Fuoss-Onsager equation and cite them if you like. And this equation still has the advantage that is reduces to Kolrausch's law for $\beta_2 = \beta_3 = 0$. So if that seems better to you than the expansion in $\sqrt{c}$, try that one.