### Apache Airflow version

2.6.3

### What happened

When running multiple standalone DAG processors with separate DAG directories using the `--subdir` argument the processors keep deactivating each other's DAGs (and reactivating their own).

After stepping through the code with a debugger I think the issue is that the calls [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L794) and [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L798) have no awareness of the DAG directories.

### What you think should happen instead

The DAG processors should not touch each other's DAGs in the metadata DB.

### How to reproduce

Start two or more standalone DAG processors with separate DAG directories and observe (e.g. via the UI) how the list of active DAGs keeps changing constantly.

### Operating System

Linux 94b223524983 6.1.32-0-virt #1-Alpine SMP PREEMPT_DYNAMIC Mon, 05 Jun 2023 09:39:09 +0000 x86_64 x86_64 x86_64 GNU/Linux

### Versions of Apache Airflow Providers

_No response_

### Deployment

Other Docker-based deployment

### Deployment details

_No response_

### Anything else

_No response_

### Are you willing to submit PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

warnings.warn(
            "`DAG.is_fixed_time_schedule()` is deprecated.",
            category=RemovedInAirflow3Warning,
            stacklevel=2,
        )
        try:
            return not self.timetable._should_fix_dst
        except AttributeError:
            return True
    def following_schedule(self, dttm):
        """
        Calculate the following schedule for this dag in UTC.
        :param dttm: utc datetime
        :return: utc datetime
        """
        warnings.warn(
            "`DAG.following_schedule()` is deprecated. Use `DAG.next_dagrun_info(restricted=False)` instead.",
            category=RemovedInAirflow3Warning,
            stacklevel=2,
        )
        data_interval = self.infer_automated_data_interval(timezone.coerce_datetime(dttm))
        next_info = self.next_dagrun_info(data_interval, restricted=False)
        if next_info is None:
            return None
        return next_info.data_interval.start
    def previous_schedule(self, dttm):