The answer is no - the point is that finitely generated projective modules are locally free but not necessarily globally so.
For instance take a Dedekind domain $A$ which does not have unique factorization and consider a non-principal prime ideal $P$. Then $\tilde{P}_x \cong \mathcal{O}_x$ for any $x\in Spec A$ but it is not isomorphic to $A$.
This will occur for any scheme with non-trivial Picard group, in the sense that there will be line bundles (i.e. locally free sheaves of rank 1) which are not trivial.
Although if you ask that $Spec A$ be Hausdorff you probably do get the result - this is because a Hausdorff scheme is necessarily 0 dimensional. The correct notion corresponding to Hausdorff is separated and all affine schemes are separated so adding this doesn't help.