A new approach to building models of generalized parton distributions (GPDs) is discussed that is based on the factorized DD (double distribution) Ansatz within the single-DD formalism. The latter was not used before, because reconstructing GPDs from the forward limit one should start in this case with a very singular function f(β)/β rather than with the usual parton density f(β). This results in a non-integrable singularity at β=0 exaggerated by the fact that f(β)'s, on their own, have a singular β^{-a} Regge behavior for small β. It is shown that the singularity is regulated within the GPD model of Szczepaniak et al., in which the Regge behavior is implanted through a subtracted dispersion relation for the hadron-parton scattering amplitude. It is demonstrated that using proper softening of the quark-hadron vertices in the regions of large parton virtualities results in model GPDs H(x,ξ) that are finite and continuous at the "border point" x=ξ. Using a simple input forward distribution, we illustrate implementation of the new approach for explicit construction of model GPDs. As a further development, a more general method of regulating the β=0 singularities is proposed that is based on the separation of the initial single DD f(β, α) into the "plus" part [f(β,α)]_{+} and the D-term. It is demonstrated that the "DD+D" separation method allows to (re)derive GPD sum rules that relate the difference between the forward distribution f(x)=H(x,0) and the border function H(x,x) with the D-term function D(α).

19 pages, 13 figures; references added, typos corrected