I was wondering is it recommended that I take a subset of genes that are significant in one of my differential expression analyses and then try to replicate those findings in another cohort of similar demographics (both are from gene expression using cord blood right after birth) with covariates being the same. I have a feeling that the answer is no but I wanted to still check. Thanks for the insights.

No. Do the second full analysis, then you can examine the subset of samples or genes you are most interested in.