### 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_
try:
            response: Integration = call_moto(context)
        except CommonServiceException as e:
            
            if e.code == "NotFoundException":
                raise NotFoundException("Invalid Integration identifier specified")
            raise
        if integration_responses := response.get("integrationResponses"):
            for integration_response in integration_responses.values():
                remove_empty_attributes_from_integration_response(integration_response)
        return response
    def put_integration(
        self, context: RequestContext, request: PutIntegrationRequest
    ) -> Integration:
        if request.get("type") == IntegrationType.AWS_PROXY:
            integration_uri = request.get("uri") or ""
            if ":lambda:" not in integration_uri and ":firehose:" not in integration_uri:
                raise BadRequestException(
                    "Integrations of type 'AWS_PROXY' currently only supports "
                    "Lambda function and Firehose stream invocations."
                )
        moto_request = copy.copy(request)
        moto_request.setdefault("passthroughBehavior", "WHEN_NO_MATCH")
        moto_request.setdefault("timeoutInMillis", 29000)
        if request.get("type") in (IntegrationType.HTTP, IntegrationType.HTTP_PROXY):
            moto_request.setdefault("connectionType", ConnectionType.INTERNET)
        response = call_moto_with_request(context, moto_request)
        remove_empty_attributes_from_integration(integration=response)
        return response
    def delete_integration(