Marking a pool member down and drain connection by looking for a file

You are likely looking for the built in LTM SMB monitor.

There are also LTM external monitors in which you can use your own custom scripted tools (curl/ftp/API calls) to verify file existence.

As for the action to take upon the chosen monitors failure - this is handled by the action on down setting on the pool described in the below white paper. I believe the default behavior of none will suit your purposes but take a look:

https://support.f5.com/csp/article/K15095