### 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_
ApiGatewayIntegrationError,
    DynamoDBIntegration,
    EventBridgeIntegration,
    HTTPIntegration,
    KinesisIntegration,
    LambdaIntegration,
    LambdaProxyIntegration,
    MockIntegration,
    S3Integration,
    SNSIntegration,
    SQSIntegration,
    StepFunctionIntegration,
)
from localstack.services.apigateway.models import ApiGatewayStore
from localstack.utils.aws import aws_stack
LOG = logging.getLogger(__name__)
class AuthorizationError(Exception):
    pass
class RequestValidator:
    __slots__ = ["context", "rest_api_container"]
    def __init__(self, context: ApiInvocationContext, store: ApiGatewayStore = None):
        self.context = context
        store = store or get_apigateway_store(
            account_id=context.account_id, region=context.region_name
        )
        if not (container := store.rest_apis.get(context.api_id)):
            
            raise NotFound()
        self.rest_api_container = container
    def is_request_valid(self) -> bool: