Hi Lei, I think your second statement is false, even in char. $0$: Let $X=\mathbb{A}_k^1$ for some field $k$, and $U:=X\setminus {0}$. Denote the open immersion by $i$. Then $i_*\mathcal{O}_U$ is $\mathcal{O}_X$-quasi-coherent and not coherent. Consider the canonical connection on $i_*\mathcal{O}_U$: If $x$ is a coordinate on $\mathbb{A}^1_k$, then $\nabla(x):=dx$. This is a connection on $i_*\mathcal{O}_U$: if we plug in $\frac{1}{x}$ we get $-\frac{1}{x^2}dx\in i_*\mathcal{O}_U\otimes \Omega^1_{X/k}$. But the section $\frac{1}{x}$ is not contained in a $\mathcal{O}_X$-coherent sub $D_X$-module: Let $\partial$ be the operator $\partial/\partial x$, then the The smallest sub $D_X$-module containing $\frac{1}{x}$ contains $\frac{1}{x^n}$ for all $n\in \mathbb{Z}$. This is not finitely generated over $\mathcal{O}_X$.
What is true however, is that on a smooth variety any $\mathcal{O}_X$-quasi-coherent $D_X$-module is the union of its $D_X$-coherent submodules, but these do not need to be $\mathcal{O}_X$-coherent. A reference for this is this:
D-Modules, Perverse Sheaves, and Representation Theory: Cor. 1.4.17
