  We study prospects of constraining the primordial magnetic field (PMF) and
its evolution during the dark ages and cosmic dawn in light of EDGES 21-cm
signal. Our analysis has been carried out on a `colder IGM' background which is
one of the promising avenues to interpret the EDGES signal. We consider the
dark matter-baryon interactions for the excess cooling. We find that the colder
IGM suppresses both the residual free electron fraction and the coupling
coefficient between the ionised and neutral components. The Compton heating
also gets affected in colder IGM background. Consequently, the IGM heating rate
due to the PMF enhances compared to the standard scenario. Thus, a significant
fraction of the magnetic energy, for $B_0 \lesssim 0.5 \, {\rm nG}$, gets
transferred to the IGM and the magnetic field decays at a much faster rate
compared to the simple $(1+z)^2$ scaling during the dark ages and cosmic dawn.
This low PMF is an unlikely candidate for explaining the rise of the EDGES
absorption signal at lower redshift. We also see that the PMF and DM-baryon
interaction together introduces a plateau-like feature in the redshift
evolution of the IGM temperature. We find that the upper limit on the PMF
depends on the underlying DM-baryon interaction. Higher PMF can be allowed when
the interaction cross-section is higher and/or the DM particle mass is lower.
Our study shows that the PMF with $B_0$ up to $\sim 0.4 \, {\rm nG}$, which is
ruled out in the standard model, can be allowed if DM-baryon interaction with
suitable cross-section and DM mass is considered.
