$\DeclareMathOperator{\F}{\mathcal{F}}\DeclareMathOperator{\o}{\mathcal{O}}$Let $X$ be a reduced, pure dimensional, projective curve over some field $k$. Let $\F$ be a coherent and torsion-free $\o_X$-module. Let $D$ be a non-zero effective Cartier divisor on $X$ with regular global section $s$ which defines the embedding $\F \stackrel{\cdot s}{\to} \F \otimes \o_X(-D)$. We write $\F(-D) := \F \otimes_{\o_X} \o_X(-D)$.
Do we have $$\dim_k H^0(X,\F(-D)/\F) = \deg_k D\quad ?$$
This is obviously true for invertible $\F$. But what about the more general case of coherent and torsion-free sheaves?