Let me put this here for the sake of clarity. As was noted by Emerton in a comment above, this answer to a related question shows that the answer is no, for an injective $R$-module $I$, the sheaf $widetilde{I}$ is not necessarily an injective sheaf.
So if you like that answer, I suggest you click on the link above and upvote that answer.
The reference provided in that answer is to the following:
MR0617087 (82i:13013)
Dade, Everett C.
Localization of injective modules.
J. Algebra 69 (1981), no. 2, 416--425.
Localization of modules over a commutative ring $R$ with respect to a multiplicatively closed subset $S$ of $R$ is an exact functor with a large number of properties, some of which are listed in Theorem 3.76 of J. J. Rotman's book [An introduction to homological algebra, Academic Press, New York, 1979; MR0538169 (80k:18001)]. The fifth property, namely: (LI) the localization $S^{-1}E$ of any injective $R$-module $E$ is an injective $S^{-1}R$-module, is false. Two examples are given here showing that arbitrary $R$ and $S$ need not have the property (LI). Also a positive result is given, showing that (LI) holds for certain non-Noetherian $R$ and certain $S$. In particular, if $R$ is the polynomial ring $k[x_1,x_2,cdots]$ in a countable number of $x_n$ over a nonzero Noetherian ring $k$, then (LI) holds for all choices of $S$.
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