### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

APiGateway Model that being created by CDK project stores the JSON schema with a single quotes instead of double quotes which leading to an error in parsing this json object. 

you can refere to this article to see the steps 

https://dev.to/dvddpl/reduce-lambda-invocations-and-code-boilerplate-by-adding-a-json-schema-validator-to-your-api-gateway-15af

and it works fine on AWS account 

also you can look on the attached image how localstack stores the JSON object with a single quotes, I tried to edit it manually through UI but it return it back with single quotes

![Screen Shot 2022-10-18 at 4 28 08 PM](https://user-images.githubusercontent.com/106184681/196769273-0834865a-2fb3-4efc-b3a9-9da6b2ad511d.png)

### Expected Behavior

store the json schema in correct JSON format

### How are you starting LocalStack?

With the `localstack` script

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

    localstack start

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

    cdklocal deploy


### Environment

```markdown
- OS:OSX
- LocalStack: 1.0.4
```


### Anything else?

_No response_
self, context: RequestContext, rest_api_id: String, resource_id: String
    ) -> None:
        moto_rest_api = get_moto_rest_api(context, rest_api_id)
        moto_resource: MotoResource = moto_rest_api.resources.pop(resource_id, None)
        if not moto_resource:
            raise NotFoundException("Invalid Resource identifier specified")
        store = get_apigateway_store(account_id=context.account_id, region=context.region)
        rest_api = store.rest_apis.get(rest_api_id)
        api_resources = rest_api.resource_children
        
        def _delete_children(resource_to_delete: str):
            children = api_resources.get(resource_to_delete, [])
            for child in children:
                moto_rest_api.resources.pop(child)
                _delete_children(child)
            api_resources.pop(resource_to_delete, None)
        _delete_children(resource_id)
        
        parent_id = moto_resource.parent_id
        api_resources[parent_id].remove(resource_id)
    def update_resource(