  Fix $n \geq 2$ an integer, and $F$ be a totally real number field. We reduce
the shifted convolution problem for $L$-function coefficients of
$\operatorname{GL}_n({\bf{A}}_F)$-automorphic forms to the better-understood
setting of $\operatorname{GL}_2({\bf{A}}_F)$. The key idea behind this
reduction is to use the classical projection operator $\mathbb P^n_1 \varphi$
together with properties of its Fourier-Whittaker expansion. This allows us to
derive novel integral presentations for the shifted convolution problem as
Fourier-Whittaker coefficients of certain $L^2$-automorphic forms on the
mirabolic subgroup $P_2({\bf{A}}_F)$ of $\operatorname{GL}_2({\bf{A}}_F)$ or
its two-fold metaplectic cover $\overline{P}_2({\bf{A}}_F)$. We then construct
liftings of these mirabolic forms to $\operatorname{GL}_2({\bf{A}}_F)$ and its
two-fold metaplectic cover $\overline{G}({\bf{A}}_F)$ to justify expanding the
underlying forms into linear combinations of Poincar\'e series. Decomposing
each of the Poincar\'e series spectrally then allows us to derive completely
new bounds for the shifted convolution problem in dimensions $n \geq 3$. As an
application, we derive a uniform subconvexity bound for
$\operatorname{GL}_n({\bf{A}}_F)$-automorphic $L$-functions twisted by Hecke
characters. This uniform level-aspect subconvexity estimate appears to the the
first of its kind for dimensions $n \geq 3$.
