= Solution
The pointwise inequality $\widehat g(X,X)\geq g(X,X)$ implies
$$
L_{\widehat g}(c)\geq L_g(c)
\quad\hbox{and hence}\quad
d_{\widehat g}(x,y)\geq d_g(x,y).
$$
Every $d_{\widehat g}$-<Cauchy sequence> $(x_j)$ is therefore $d_g$-Cauchy. Since $g$ is <geodesic completeness>[geodesically complete], the <Hopf-Rinow theorem> makes $(M,d_g)$ a <complete metric space>, so $x_j\to x$ in $d_g$ for some $x\in M$.
On a coordinate neighbourhood with compact closure around $x$, smooth positive-definite <Riemannian metric>[Riemannian metrics] are uniformly equivalent. Thus there is $C>0$ such that
$$
\widehat g(X,X)\leq Cg(X,X)
$$
there. For all sufficiently large $j$, a short $g$-geodesic from $x$ to $x_j$ stays in this neighbourhood, and hence
$$
d_{\widehat g}(x_j,x)\leq\sqrt C\,d_g(x_j,x)\longrightarrow0.
$$
Thus $(M,d_{\widehat g})$ is complete. Another application of Hopf-Rinow shows that $\widehat g$ is geodesically complete.
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