For upgrade server 2003 Std to 2019

It isn't supported to introduce 2019 domain controllers to a 2003 DFL domain.

https://docs.microsoft.com/en-us/windows-server/identity/ad-ds/active-directory-functional-levels

Also FRS needs to be migrated to DFSR before adding 2019 domain controllers.

https://techcommunity.microsoft.com/t5/Storage-at-Microsoft/Streamlined-Migration-of-FRS-to-DFSR-SYSVOL/ba-p/425405

So you'll need to do this two-step, first by migrating to 2012 R2 or 2016, removing the 2003 DCs in order to raise DFL to minimum or 2008 R2, then do the FRS-->DFSR migration, then bring in the first 2019 domain controller.

To the domain controller migration general steps below;

I'd use dcdiag / repadmin tools to verify health correcting all errors found before starting. Then stand up the new guest, patch it fully, license it (you can use evaluations for the interim step), join existing domain, add active directory domain services, promote it also making it a GC (recommended), transfer FSMO roles over (optional), transfer pdc emulator role (optional), use dcdiag / repadmin tools to verify health, when all is good you can decommission / demote old one, move on to next one.