we have migrated from ec2 and embedded DB to eks with external DB (another VPC).
So there is no difference from the app side about VPC, just make sure you have connections. And create backup, restore to new DB (either embedded or RDS).
Also that could help:
I guess there is actually no difference between restoring DB dump to either external or internal DB. Just make sure to dump and restore it properly.
Also, I would recommend not to stop old instance until you check your new instance with old data.