In FGA, no.232, Thm 3.1, Grothendieck shows that if $f: X\to S$ is flat, projective and finitely presented, with reduced and irreducible geometric fibers, then $\operatorname{Pic}_{X/S}$ is representable by a separated $S$-scheme, locally of finite presentation over $S$.
As for smoothness, unfortunately the "well-known" result you cite is actually not true in general--in particular, Igusa constructed a smooth projective surface (here is the original paper, written in somewhat archaic language) in characteristic $2$ with non-reduced Picard scheme; the example is the quotient of a particular Abelian surface by a fixed-point-free involution. The smoothness result you claim is true in characteristic zero.
Luckily, Theorem 5 of Section 8.4 of Bosch-Lütkebohmert-Raynaud's Neron Models [BLR] answers your question when $X$ is an Abelian $S$-scheme; in this case the Picard functor is representable by another Abelian $S$-scheme, and is in particular smooth.
Just a warning: in the general case (e.g. if $f: X\to S$ has no section, for example), the Picard functor represented might not be what you think it is, but rather the etale or fppf sheafification of the usual Picard functor. (This can happen e.g. if $X$ has no rational points and $S$ is $\operatorname{Spec}(k)$.) Of course an Abelian scheme always admits a section, so you're OK in this situation.
BTW, Chapter 8 of BLR is a great place to learn about this stuff.
