I find it kind of counter intuitive, that the result of an advection gets more smeared out at the borders when decreasing the timestep (which should make it more accurate).
Let there be a equally spaces grid $x_1, \dots, x_n$ with a constant 1D advection $v=1$. Let the initial values be $$u^0=[0, 0, 1, 1, 0, 0]$$
with the corresponding divergence field
$$\nabla\cdot (uv) = [0,0,1,0,-1,0]$$
With a timestep of $\delta_t = 1$ we get for one timestep with the explicit euler forward time scheme and forward differences for the derivative:
$$u^1=[0, 0, 0, 1, 1, 0]$$
But with a smaller timestep $\delta_t=0.5$ we get:
$$ u^{0.5}=\left[0,0,\frac{1}{2}, 1, \frac{1}{2}, 0\right]\\ u^1=\left[0,0,\frac{1}{4}, \frac{1}{2}, \frac{1}{2}, \frac{1}{4}\right] $$
So while the CFL number requires a smaller timestep for stability, in this case a smaller number leads to unsharp boundaries and decreasing the $\delta_x$ helps.
Is there some other stablity condition than CFL and Peclet numbers for this problem?