= Solution
At each wavevector the orbital-basis <Hamiltonian> is
$$
H(\mathbf k)=\begin{pmatrix}E_p(k)&\alpha\\\alpha&E_{\rm fl}\end{pmatrix}.
$$
A phase choice makes the momentum-independent mixing real and nonnegative. The <characteristic equation> is
$$
(E_p-E)(E_{\rm fl}-E)-\alpha^2=0,
$$
so
$$
\boxed{E_\pm(k)=\frac12\left[E_p+E_{\rm fl}\pm\sqrt{(E_p-E_{\rm fl})^2+4\alpha^2}\right].}
$$
For $\alpha>0$, an eigenvector obeys $c_{\rm fl}/c_p=(E_\pm-E_p)/\alpha$. At the bare crossing, the normalized states are $(|p\rangle\pm|{\rm fl}\rangle)/\sqrt2$ with energies $E_{\rm fl}\pm\alpha$. The lower state's parabolic-orbital weight is
$$
|c_{p,-}|^2=\frac12\left[1-\frac{E_p-E_{\rm fl}}{\sqrt{(E_p-E_{\rm fl})^2+4\alpha^2}}\right],
$$
which tends to one below the crossing and zero above it. This makes the exchange of band character in the sketches explicit.
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