Calculate stable time step DG method for advection-diffusion

For stable time steps for the RKDG method for transport equations we require that

$$
Delta t le frac{Delta x CFL}{(2k + 1)|lambda|},
$$
where $lambda$ is the eigenvalue of our conservation law and $k = 0, 1, dots$. For diffusion I believe we require that

$$
Delta t le frac{Delta x^{2}}{nu},
$$
where $nu$ is the diffffusion coefficient. To calculate a stable time step I am doing the following,
$$
Delta t le min left{frac{Delta x^{2}}{nu},frac{Delta x CFL}{(2k + 1)|lambda|}right}.
$$
It works reasonably well for $k = 1$ up to 160 elements. For $k = 2$, it only produces stable time steps for up to 80 elements. The solution does not blow up but I do not get the correct rate of convergence. As such, I was curious if someone had a literature reference or could provide the correct expression on how to calculate stable time steps that would yield the correct rates of convergence. For the time being I would like to stick with explicit RK methods for simplicity as I’m still learning DG. As a side note, the CFL condition I’m choosing is quite small, i.e. $CFL = 0.05$ to $CFL = 0.01$.