  Nielsen-Olesen vortices in the Abelian-Higgs (AH) model are the simplest
realisations of cosmic strings in a gauge field theory. Large-scale numerical
solutions show that the dominant decay channel of a network of AH strings
produced from random initial conditions is classical field radiation. However,
they also show that with special initial conditions, loops of string can be
created for which classical field radiation is suppressed, and which behave
like Nambu-Goto (NG) strings with a dominant decay channel into gravitational
radiation. This indicates that cosmic strings are generically sources of both
high-energy particles and gravitational waves. Here we adopt a simple
parametrisation of the AH string network allowing for both particle and
gravitational wave production. With a reference to a specific model for NG-like
loop distribution, this sets the basis for a ``multi-messenger'' investigation
of this model. We find that, in order to explain the NANOGrav detection of a
possible gravitational wave background, while satisfying the constraint on
NG-like loop production from simulations and bounds from the cosmic microwave
background, the tension of the AH string in Planck units $G\mu$ and the
fraction of the NG-like loops $f_{\rm NG}$ should satisfy $G\mu f_{\rm
NG}^{2.6} \gtrsim 3.2\times 10^{-13}$ at 95$\%$ confidence. On the other hand,
for such string tensions, constraints from the diffuse gamma-ray background
(DGRB) indicate that more than 97$\%$ of the total network energy should be
converted to dark matter (DM) or dark radiation. We also consider joint
constraints on the annihilation cross-section, the mass, and the relic
abundance of DM produced by decays of strings. For example, for a DM mass of
500 GeV, the observed relic abundance can be explained by decaying AH strings
that also account for the NANOGrav signal.
