There is the "flabbification" or "flasquification" functor used in the Godement resolution. Namely, given a sheaf $\mathcal{F}$ on $X$, let $\mathcal{F}_x$ be the stalk at a point $x$. Then we define the sheaf $\Phi(\mathcal{F})$ to have sections $$\Gamma(U, \Phi(\mathcal{F}))=\prod_{x\in U} \mathcal{F}_x$$ with the obvious restriction maps. (This is the same as endowing the etale space of $\mathcal{F}$ with the trivial topology and considering the sheaf of sections.) There is a natural injection $\mathcal{F}\to \Phi(\mathcal{F})$ sending a section of $\mathcal{F}$ to its stalks. Note that $\Phi$ is functorial; indeed, it is the right adjoint to the natural inclusion of the full subcategory (Flasque sheaves on $X$) $\hookrightarrow$ (Sheaves on $X$).
See for example this Wikipedia article or Godement's book on sheaf theory.
Edit: One can also mimic your construction of $C^k_{nd}$ as follows. Let $\mathcal{F}_{nd}$ have global sections given by $\bigcup \Gamma(X-\partial U, \mathcal{F})/\sim$ where the union is taken over all open sets $U$, and we say $(f, X-\partial U)\sim (f', X-\partial U')$ if $f=f'$ when restricted to $X-(\partial U\cup \partial U')$. Then local sections will be restrictions of these global sections. This will always be flabby, but will not necessarily have a morphism $\mathcal{F}\to \mathcal{F}_{nd}$ unless $\mathcal{F}$ has enough sections (for example, if $\mathcal{F}$ is fine, as in your example).