  We propose a new method for fitting the full-shape of the Lyman-$\alpha$
(Ly$\alpha$) forest three-dimensional (3D) correlation function in order to
measure the Alcock-Paczynski (AP) effect. Our method preserves the robustness
of baryon acoustic oscillations (BAO) analyses when it comes to measuring the
position of the acoustic peak, while also providing extra cosmological
information from a broader range of scales. We compute forecasts for the Dark
Energy Spectroscopic Instrument (DESI) using the Ly$\alpha$ auto-correlation
and its cross-correlation with quasars, and show how this type of analysis
improves cosmological constraints. The DESI Ly$\alpha$ BAO analysis is expected
to measure $H(z_\mathrm{eff})r_\mathrm{d}$ and
$D_\mathrm{M}(z_\mathrm{eff})/r_\mathrm{d}$ with a precision of $\sim0.9\%$
each, where $H$ is the Hubble parameter, $r_\mathrm{d}$ is the comoving BAO
scale, $D_\mathrm{M}$ is the comoving angular diameter distance and the
effective redshift of the measurement is $z_\mathrm{eff}\simeq2.3$. By fitting
the AP parameter from the full shape of the two correlations, we show that we
can obtain a precision of $\sim0.5-0.6\%$ on each of
$H(z_\mathrm{eff})r_\mathrm{d}$ and
$D_\mathrm{M}(z_\mathrm{eff})/r_\mathrm{d}$. Furthermore, we show that a joint
full-shape analysis of the Ly$\alpha$ auto-correlation and its
cross-correlation with quasars can measure the linear growth rate times the
amplitude of matter fluctuations on scales of $8\;h^{-1}$Mpc,
$f\sigma_8(z_\mathrm{eff})$. Such an analysis could provide the first ever
measurement of $f\sigma_8(z_\mathrm{eff})$ at redshift $z_\mathrm{eff}>2$. By
combining this with the quasar auto-correlation in a joint analysis of the
three high-redshift two-point correlation functions, we show that DESI will be
able to measure $f\sigma_8(z_\mathrm{eff}\simeq2.3)$ with a precision of
$5-12\%$, depending on the smallest scale fitted.
