  The supersymmetrized DFSZ axion model is especially compelling in that it
contains 1. the SUSY solution to the gauge hierarchy problem, 2. the
Peccei-Quinn (PQ) solution to the strong CP problem and 3. the Kim-Nilles
solution to the SUSY mu problem. In a string setting, where a discrete
R-symmetry ({\bf Z}_{24}^R for example) may emerge from the compactification
process, a high-quality accidental axion (accion) can emerge from the
accidental, approximate remnant global U(1)_{PQ} symmetry where the decay
constant f_a is linked to the SUSY breaking scale, and is within the
cosmological sweet zone. In this setup, one also expects the presence of
stringy remnant moduli fields \phi_i. Here, we consider the situation of a
single light modulus \phi coupled to the PQMSSM in the early universe, with
mixed axion plus higgsino-like WIMP dark matter. We evaluate dark matter and
dark radiation production via nine coupled Boltzmann equations and assess the
severity of the cosmological moduli problem (CMP) along with dark matter and
dark radiation production rates. We find that typically the light modulus mass
should be m_{\phi}>~ 10^4 TeV to avoid the moduli-induced dark matter
overproduction problem. If one is able to (anthropically) tune the modulus
field amplitude, we find a value of \phi_0 <~ 10^{-7}m_P would be required to
solve the overall CMP.
