### 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_
if not integration:
        return integration
    if not integration.get("integrationResponses"):
        integration.pop("integrationResponses", None)
    if integration.get("requestParameters") is None:
        integration.pop("requestParameters", None)
    return integration
def remove_empty_attributes_from_model(model: Model) -> Model:
    if not model.get("description"):
        model.pop("description", None)
    return model
def remove_empty_attributes_from_integration_response(integration_response: IntegrationResponse):
    if integration_response.get("responseTemplates") is None:
        integration_response.pop("responseTemplates", None)
    return integration_response
def validate_model_in_use(moto_rest_api: MotoRestAPI, model_name: str) -> None:
    for resource in moto_rest_api.resources.values():
        for method in resource.resource_methods.values():
            if method.request_models and model_name in set(method.request_models.values()):
                path = f"{resource.get_path()}/{method.http_method}"
                raise ConflictException(
                    f"Cannot delete model '{model_name}', is referenced in method request: {path}"
                )
def create_custom_context(