In addition to the other advice you've received:

• Put the code to prompt for a user name in a common subroutine. You can always pass a descriptive phrase ('to add', 'to delete') as a parameter.
• Open DB.txt once at the start of your code, and (if necessary) back up the old file and write the new file once at the end of your program (preferably in a subroutine). This enables you to remove multiple FH names.