Replacing server 2003 with Server 2019

This will be a two-step process.

The prerequisite before introducing the first 2016 domain controller: domain functional level needs to be 2003 or higher

I'd use dcdiag / repadmin tools to verify health `correcting all errors found` before starting `any` operations. Then stand up the new 2019 or 2022, patch it fully, license it, 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 again verify health, when all is good you can decommission / demote old one.
 

The two prerequisites to introducing the first 2019 or 2022 domain controller are that domain functional level needs to be 2008 or higher and older sysvol FRS replication needs to have been migrated to DFSR
https://techcommunity.microsoft.com/t5/Storage-at-Microsoft/Streamlined-Migration-of-FRS-to-DFSR-SYSVOL/ba-p/425405 

Once done, repeat above but installing Server 2019