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.
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