  We study the inclusive and exclusive cross sections of $e^+e^-\to
\text{hadrons}$ for center-of-mass energies between 3.70 GeV and 3.83 GeV to
infer the mass, width, and couplings of the $\psi(3770)$ resonance. By using a
coupled-channel K-matrix approach, we setup our analysis to respect unitarity
and the analyticity properties of the underlying scattering amplitudes. We fit
several models to the full dataset and identify our nominal results through a
statistical model comparison. We find that, accounting for the interplay
between the $\psi(2S)$ and the $\psi(3770)$, no further pole is required to
describe the $\psi(3770)$ line shape. In particular we derive from the pole
location $M_{\psi(3770)} = 3778.8 \pm 0.3$ MeV and $\Gamma_{\psi(3770)} = 25.0
\pm 0.5$ MeV. Moreover, we find the decay to $D^+D^-$ and $D^0\bar{D}^0$ to be
consistent with isospin symmetry and derive an upper bound on the branching
ratio $\mathcal{B}(\psi(3770) \to \textrm{non-}D\bar{D}) < 6\%$ at $90\%$
probability.
