Adjoint of the covariant derivative on a Riemannian manifold
You can explicitly compute the adjoint by integrating by parts: the metric-compatibility of $\nabla$ gives $$ \begin{align} g(\nabla_X \alpha, \beta) &= X g(\alpha,\beta) - g(\alpha, \nabla_X \beta) \\ &=\text{div}(g(\alpha,\beta)X)-g(\alpha,\beta)\text{div}(X)-g(\alpha,\nabla_X \beta) \end{align}$$
and thus integrating over a region containing the supports of $\alpha$ and $\beta$ you get
$$\langle \nabla_X \alpha, \beta \rangle = \langle\alpha,-\text{div}(X) \beta-\nabla_X\beta\rangle$$
so $\nabla_X^* = -\text{div}(X) - \nabla_X$.