### 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_
ApiName:
    Type: String
  IntegrationUri:
    Type: String
Resources:
  Api:
    Type: AWS::Serverless::Api
    Properties:
      StageName: dev
      Name: !Ref ApiName
      DefinitionBody:
        swagger: 2.0
        info:
          version: "1.0"
          title: "Public API"
        basePath: /base
        schemes:
        - "https"
        x-amazon-apigateway-binary-media-types:
        - "*/*"
        paths:
          /test:
            post:
              responses: {}
              x-amazon-apigateway-integration:
                uri: !Ref IntegrationUri
                httpMethod: "POST"
                type: "http_proxy"
"""
def test_cfn_apigateway_aws_integration(deploy_cfn_template, aws_client):