How do I "convert" the newly added teams to this methodology and way of thinking with as little trauma and resistance from the members of these teams?
Actually start -- right now -- with backlog definition, prioritization and sprint planning.
They won't "get it" right away. That doesn't matter -- explanations don't really help as much as simply doing it.
Get them started with daily standups today. Immediately. No waiting around. No "conversion". Just start.
Agree to just one sprint to get started with a reasonable scope of work. They'll completely mess it up, but that's okay.
Redefining the sprint's objectives -- rearranging the backlog -- is easiest done hands-on with real deliverables in a real project with real sprint deadlines approaching.
Moving from long, long coding followed by a burst of insane testing to shorter sprints isn't really all that hard. Most folks prefer it. We mostly do waterfall to placate managers and accountants.
Do a retrospective and continue the cycle allowing them to make their own improvements.
The best way to learn Agile methods is to just start doing Agile methods.
As questions come up ("How can we prioritize if we don't know every single possible requirement in advance?") address those specific questions with advice based on the specific backlog and sprint you have to tackle right now.
Avoid hypothetical answers. Just get started.
Ideally, part way through the second sprint, folks will say that this "Agile thing" isn't all that interesting. They always did this. They just didn't call it Agile.