How to migrate the specific Linux partition rather than the entire disk

Unfortunately you can only migrate an entire disk. MGN has no partition awareness so it is not possible to resize a disk during migration. You can change the disk type (eg io1 to gp3 etc). See this page for the settings which can be changed during migration.

In this scenario I usually advise customers to explore RePlatform options because the cost of lift and shift has become higher due to the overprovision. For example can you redeploy the application on a new optimized instance with latest OS potentially using infrastucture as code techniques. You need to evaluate if this engineering effort is equal to the cost savings compared to the lift and shift