EDIT: My original question was very badly posed, and has been edited substantially.
Suppose $ D' $ is a prime divisor on $ X $ containing a non-singular point $ x, $ and $ f \in I(D') \subset \mathcal{O}_{X,x}. $ Then $ f = up_{1}\dots p_{k}, $ where $ u $ is a unit in $ \mathcal{O}_{X,x} $ and $ p_{i} $ is an irreducible polynomial for $ i \in \lbrace 1,\dots, k \rbrace. $ After a bit of work, we find that $ I(D') = (p_{i}) $ for some $ i. $
So every prime divisor on $ X $ containing $ x $ is locally principal. Suppose $ \lbrace D'_{i} \rbrace_{i=1} $ is some family of such prime divisors. I would like to infer that any Weil divisor $ D = \sum_{i}^{n} n_{i}D'_{i} $ is also locally principal based of the fact that we know the $ D'_{i}$s are.
ASIDE: I am aware of the following fact about Weil divisors on $ \text{Spec}(\mathcal{O}_{X,x}): $ if $ X $ is a variety, and $ \mathcal{O}_{X,x} $ is a UFD for every nonsingular point $ x \in X, $ then any Weil divisor $ D \ni x $ on $ X $ induces a Weil divisor $ D_{x} $ on $ \text{Spec}(\mathcal{O}_{X,x}), $ and $ D_{x} $ is principal.