Yes. You can add an existing student (a student who has already created an account) to a class section simply by clicking on the "Manage Students" tab from the left-navigation menu and then clicking the "Add/Move Existing Student to Class" button in the upper-right corner.
Alternatively, you can simply provide the student with the class code of the new class section and instruct the student to click on the profile button in the upper-right corner of her student dashboard and then enter the class code into the class section field to join the new class.
All of the student's scores (with the exception of the diagnostic assessment scores) will remain with the student and appear in the Progress Reports of the new class section.
If you have synced your GrammarFlip account with your Google Classroom, any classroom management (ie, removing/adding students) should be done via Google Classroom (which is why the "Manage Students" tab is now missing from your left-navigation menu). First make any changes to your student roster in your Google Classroom, and then you can click the "Sync with Classroom" button on your GrammarFlip dashboard. The changes you made in Google Classroom will then be reflected in your GrammarFlip dashboard.