  We explore the effects of the expected higher cosmic ray (CR) ionization
rates $\zeta_{\rm CR}$ on the abundances of carbon monoxide (CO), atomic carbon
(C), and ionized carbon (C$^+$) in the H$_2$ clouds of star-forming galaxies.
The study of Bisbas et al. (2015) is expanded by: a) using realistic
inhomogeneous Giant Molecular Cloud (GMC) structures, b) a detailed chemical
analysis behind the CR-induced destruction of CO, and c) exploring the thermal
state of CR-irradiated molecular gas. CRs permeating the interstellar medium
with $\zeta_{\rm CR}$$\gtrsim 10\times$(Galactic) are found to significantly
reduce the [CO]/[H$_2$] abundance ratios throughout the mass of a GMC. CO
rotational line imaging will then show much clumpier structures than the actual
ones. For $\zeta_{\rm CR}$$\gtrsim 100\times$(Galactic) this bias becomes
severe, limiting the utility of CO lines for recovering structural and
dynamical characteristics of H$_2$-rich galaxies throughout the Universe,
including many of the so-called Main Sequence (MS) galaxies where the bulk of
cosmic star formation occurs. Both C$^+$ and C abundances increase with rising
$\zeta_{\rm CR}$, with C remaining the most abundant of the two throughout
H$_2$ clouds, when $\zeta_{\rm CR}\sim (1-100)\times$(Galactic). C$^+$ starts
to dominate for $\zeta_{\rm CR}$$\gtrsim 10^3\times$(Galactic). The thermal
state of the gas in the inner and denser regions of GMCs is invariant with
$T_{\rm gas}\sim 10\,{\rm K}$ for $\zeta_{\rm CR}\sim (1-10)\times$(Galactic).
For $\zeta_{\rm CR}$$\sim 10^3\times$(Galactic) this is no longer the case and
$T_{\rm gas}\sim 30-50\,{\rm K}$ are reached. Finally we identify OH as the key
species whose $T_{\rm gas}-$sensitive abundance could mitigate the destruction
of CO at high temperatures.
