In this paper, we introduce a new class of Finsler metrics that generalize the well-known $(\alpha, \beta)$-metrics.
These metrics are defined by a Riemannian metric $\alpha$ and two 1-forms $\beta=b_i(x)y^i$ and $\gamma=\gamma_i(x)y^i$.
This new class of metrics not only generalizes $(\alpha, \beta)$-metrics, but also includes other important Finsler metrics,
such as all (generalized) $\gamma$-changes of generalized $(\alpha, \beta)$-metrics, $(\alpha, \beta)$-metrics, and spherically symmetric Finsler metrics in $\mathbb{R}^n$.
We find a necessary and sufficient condition for this new class of metrics to be locally projectively flat.
Furthermore, we prove the conditions under which these metrics are of Douglas type.

Keywords: Finsler geometry, $(\alpha, \beta, \gamma)$-metrics, Projectively flat, Douglas space.