### 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_
def get_apigw_resource_params(
            properties: dict, logical_resource_id: str, resource: dict, stack_name: str
        ) -> dict:
            result = {
                "restApiId": properties.get("RestApiId"),
                "pathPart": properties.get("PathPart"),
                "parentId": properties.get("ParentId"),
            }
            if not result.get("parentId"):
                
                apigw = connect_to().apigateway
                resources = apigw.get_resources(restApiId=result["restApiId"])["items"]
                root_resource = ([r for r in resources if r["path"] == "/"] or [None])[0]
                if not root_resource:
                    raise Exception(
                        "Unable to find root resource for REST API %s" % result["restApiId"]
                    )
                result["parentId"] = root_resource["id"]
            return result
        def _handle_result(result, resource_id, resources, resource_type):
            resources[resource_id]["PhysicalResourceId"] = result["id"]
        return {
            "create": {
                "function": "create_resource",
                "parameters": get_apigw_resource_params,
                "result_handler": _handle_result,
            }
        }
class GatewayMethod(GenericBaseModel):