This will need expansion by a more knowledgable person, but as memory serves, it was proved by Mayer and Mumford that the closure in Ag of the locus of traditional Jacobians is the set of products of Jacobians. This is probably exposed first in a talk in the 1964 Woods Hole talks on James Milne's site. But let us try to explain this more in dim two.
A two diml ppav is a compact 2 torus A containing a curve C carrying the homology class a1xb1 + a2xb2, where the aj,bj are a basic symplectic homology basis of H1(A).
It follows from the topological Pontrjagin product that the induced map from the Albanese variety of C to A, has topological degree one, hence is an isomorphism.
Since it also induces the identity map on C, it also preserves the polarization.
Let me speculate on the special cases. If C is reducible it is known that A is a product of elliptic curves. If C is irreducible and singular then I guess the normalization map extends to a map of the Albanese of C to A. But that seems to imply the image of C in A does not span, a contradiction.
So it seems that any curve C contained in a two diml ppav A and carrying the class of a principal polarization, is smooth and induces an isomorphism from the Albanese of the curve to the ppav.
I hope there is some useful information in this.