You have my sympathy, I have struggled with torsion-freeness of Picard and class groups too.

As Donu hinted in his comment on Damiano's answer, a torsion element of the Picard group gives you a cyclic cover of $X$, so triviality would follow from certain purity results for etale covers. A fact that is interesting (at least for me) is that one could use the purity of the local ring at the origin of the affine cone over $X$, see Lemma 9 of this paper.

(purity for the local ring here is in the sense of Grothendieck, that is to say, the restriction of etale covers over $\text{Spec} A$ to the punctured spectrum is an equivalence) .

This shows, for example, Grothendieck's classical result that if $X$ is a complete intersection of dimension at least $2$, then $\text{Pic}(X)$ is torsion-free. But Cutkosky's paper above gives quite more general results, see Theorem 19, the Cor after Theorem 26 and the Examples after Theorem 22 of his paper. Basically, his results say that if $X$ is locally a complete intersection in high enough codimension relative to the deviation of the local ring at the vertex of the cone, then $\text{Pic} (X)$ is torsion-free. These apply to a few Grasmanians and Pfaffians, and hopefully they can be helpful to what you want to do.

Also note that various results which give vanishing of $H_1$ or $\pi_1$ when $X$ is relatively small codimension subvariety of the projective space can be found in the paper of Lyubeznik and the references therein.

ADDED: I had a chance to look at Lyubeznik's paper and actually he has Theorem 11.2, which says that if $Y\subset \mathbb P^n_{\mathbb C}$ is an irreducible algebraic set of codimension $c< n/2$, then $\text{Pic}(Y)$ is torsion-free.