I think you should make your question more precise. If $X$ is quasi-projective, and $f:\widetilde X\to X$ is a projective birational morphism, then $f$ is a blow-up of an appropriate ideal sheaf. I am guessing this is not what you're after. (If you allow blow ups of fractional ideals, then you can drop even the quasi-projective assumption. Not assuming that $f$ is projective would make the question less interesting).
Also the way you phrased your question, I think there is a simple example: take any blow ups and then choose a subscheme of $X$ with the same support, but different scheme structure than the subscheme that was blown up.
Anyway, I will assume that you want an example that's not a blow up centered at a smooth subvariety.
I think it is possible to give examples satisfying property #2 relatively easily.
Let's look for $f$ as the blow up of an ideal sheaf $\mathscr I$. So $\widetilde X=\mathrm{Proj}_X(\oplus_d \mathscr I^d)$. Then the pre-image of the subvariety $Y\subset X$ defined by the ideal $\mathscr I$ is given by $\widetilde Y=\mathrm{Proj}_Y(\oplus_d \mathscr{I^d/I^{d+1}})$. Now if $X$ is Cohen-Macaulay and $Y$ is a complete intersection in $X$, (i.e., $\mathscr I$ is generated by a regular sequence), then $\mathscr{I/I^2}$ is locally free and $\mathscr{I^d/I^{d+1}}\simeq \mathrm{Sym}^d(\mathscr{I/I^2})$ and hence $\widetilde Y\simeq \mathbb P(\mathscr{I/I^2})$.
Property #3 is kind of a red herring. The $(-1)$-twist is almost automatic, it comes from the construction of the blow up of $\mathscr I$.
My example is not likely to satisfy property #1, that is $\widetilde X$ is probably always singular if $Y$ is.
Finally, here is a simple concrete example: Let $X$ be a plane (or any smooth surface) and $\mathscr I=(x^2,y^2)$ where $x,y$ are local coordinates at a point. The blow up will be the surface with a pinch point (locally around the interesting singularity defined by $x^2z=y^2$) with the singular line contracted to a point. I think it is relatively easy to check that this satisfies properties #2 and #3.
As far as finding an example that satisfies all 3 properties, I would not hold my hopes very high. If $\dim X=2$, then these properties just mean that we have a $(-1)$-curve and we know that that is always the blow up of a smooth point. In higher dimensions I would start by taking general hyperplane cuts to reduce to the case when the image is a point. Then Property #2 says that the preimage is a $\mathbb P^n$ with normal bundle $\mathscr O_{\mathbb P^n}(-1)$ so there is a good chance that one can prove that it is indeed a blow up. Then one would get (probably) that the original map is a blow up at general points and then possibly one can prove that then it has to be a blow up everywhere, but I am not entirely sure about that.
EDIT Mike Roth in the comments below gives a nice example of a blow up along a non-smooth subvariety such that the resulting variety is actually smooth. I don't think this satisfies the other conditions, but it might suggest that proving that those three conditions indeed characterize blow ups along smooth subvarieties is probably not easy.