Begin by accessing the OneSignal API documentation at https://documentation.onesignal.com/reference. You'll need to create a REST API key from your OneSignal account. Navigate to the "Settings" tab, then "API & Keys," and generate a new API key with the necessary permissions to read data.

Ensure you have a tool that can make HTTP requests and handle JSON data, such as Python with the `requests` library. If you haven't already, install Python and then run `pip install requests` in your command line to install the library.

Use your preferred programming language to authenticate and fetch data from OneSignal. For example, in Python, use the `requests` library to make a GET request to the OneSignal API endpoint. Include the API key in the header:
```python
import requests

url = "https://onesignal.com/api/v1/notifications"
headers = {
"Authorization": "Basic YOUR_API_KEY"
}
response = requests.get(url, headers=headers)
data = response.json()
```
Replace `YOUR_API_KEY` with your actual API key.

Once you have fetched the data, process it as needed. This may involve filtering out unnecessary information or transforming the data structure to fit your specific requirements. Use Python's list comprehensions and dictionary operations to manipulate the JSON data.

Ensure the processed data is in a proper JSON format. While Python dictionaries can be directly converted to JSON, you might want to structure the data in a specific way or include only certain fields. Use Python's `json` module to handle conversions:
```python
import json

processed_data = # your filtered and structured data
json_data = json.dumps(processed_data, indent=4)
```

Write the JSON data to a file on your local system. Choose a suitable file name and directory, and ensure you have write permissions for that location:
```python
with open('onesignal_data.json', 'w') as json_file:
json_file.write(json_data)
```
This will create a file named `onesignal_data.json` in the current working directory with your OneSignal data.

After saving the data, it's crucial to validate the JSON file to ensure its integrity. Use a JSON validator tool or library to check for any syntax errors. You can use Python's `json` module to load the file and verify:
```python
with open('onesignal_data.json', 'r') as json_file:
try:
data = json.load(json_file)
print("JSON file is valid.")
except json.JSONDecodeError:
print("JSON file is invalid.")
```
This step ensures that your JSON file is correctly formatted and ready for further use or distribution.

By following these steps, you can effectively move data from OneSignal to a JSON file without relying on third-party connectors or integrations.