  Let $\{X_{i}, i\in J\}$ be a family of locally dependent non-negative
integer-valued random variables with finite expectation and variance. We
consider the sum $W=\sum_{i\in J}X_i$ and establish general error upper bounds
for the total variation distance $d_{TV}(W, Y^{d})$, where $Y^{d}$ is the
discretized normal distribution. The major ingredient of the proof is to
approximate $W$ by a three-parametric intermediate random variable $M$ based on
Stein's method. As applications, we study in detail four well-known examples,
which are counting vertices of all edges point inward, birthday problem,
counting monochromatic edges in uniformly colored graphs, and triangles in the
Erd\H{o}s-R\'{e}nyi random graph. Through delicate analysis and computations we
obtain sharper upper error bounds than existing results.
