This is true. For A/mathbbC an abelian variety, L an ample line bundle on A, then any line bundle
MinoperatornamePic0(A) -- over mathbbC, this is equivalent to having first Chern class zero -- is of the form T∗xLotimesL−1 for some xinA. (e.g. Theorem 1 on p. 77 of Mumford's Abelian Varieties).
Applying this theorem with L=L(D2), M=L(D1)−L(D2), we get that
L1−L2=T∗x(L2)−L2, so
L1=T∗x(L2).
So D1 and x+D2 (meaning translation of D2 by x!) must be linearly equivalent, but by your assumption h0(L(D1))=h0(L(D2))=1, they are each the unique effective
divisors in their linear equivalence classes, so we must have D1=x+D2.
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