What sort of issues did you hit? In all honesty I'm not sure I've been more impressed by another AWS service.
We burned 3 weeks just trying to get support to provide a sensible response. I never got the sense anyone replying to us knew any more than the surface level about the infrastructure of how DMS worked.
I've used it fairly frequently over a number of years so maybe the issues we hit on the learning curve have dimmed.
We also deliberately kept things as simple as possible at first and focused on DMS's ability to move the data from source to target and really only tweaked settings that increased security or reliability. We stayed well away from any of the data transformation or more complex functionality.
We treated these carefully and tested as we made them but never had any real issues with them. From memory DMS could cope with adding columns pretty transparently. One setting we invested in configuring and understanding was to allow DMS to attempt to recover from replication failures. This allowed it to error on DDL changes and attempt to recover. This usually involved restarting the task, but it would do this transparently as part of the recovery.