### 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_
apigateway = connect_to().apigateway
            provider = cls(resource)
            props = provider.props
            integration = props.get("Integration")
            if integration:
                api_id = props["RestApiId"]
                res_id = props["ResourceId"]
                kwargs = keys_to_lower(integration)
                if uri := integration.get("Uri"):
                    
                    
                    if integration.get("Type") in ["HTTP", "HTTP_PROXY"]:
                        rs = urlparse(uri)
                        if not rs.path:
                            uri = "{}/".format(uri)
                    kwargs["uri"] = uri
                integration_responses = kwargs.pop("integrationResponses", [])
                method = props.get("HttpMethod")
                kwargs["requestParameters"] = kwargs.get("requestParameters") or {}
                kwargs["requestTemplates"] = kwargs.get("requestTemplates") or {}
                apigateway.put_integration(
                    restApiId=api_id,
                    resourceId=res_id,