Tilting Billingsley's model toward a giant prime: two phase transitions

Billingsley's theorem states that the normalized logarithms of the prime factors of a uniform random integer in $[1,x]$ converge to the Poisson--Dirichlet law $PD(1)$, while weighting an integer $n$ by the generalized divisor function $d_θ(n)$ gives $PD(θ)$. We ask what remains of this picture when