How do I know which binary log file to open for continous backup?
You need them all.  Or at least any that contain the "point in time" that you need, plus all that come after that file.

When one binlog gets "full" (according to a setting), it stops being written to, and another one starts.

FLUSH LOGS also stops the writing to one binlog and starts another.  Presumably, your full backlog does such.  That becomes the point at which you can toss all older binlogs.