### 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_
Account,
    ApigatewayApi,
    ApiKey,
    ApiKeys,
    Authorizer,
    Authorizers,
    BadRequestException,
    BasePathMapping,
    BasePathMappings,
    Blob,
    Boolean,
    ClientCertificate,
    ClientCertificates,
    ConflictException,
    ConnectionType,
    CreateAuthorizerRequest,
    CreateRestApiRequest,
    DocumentationPart,
    DocumentationPartIds,
    DocumentationPartLocation,
    DocumentationParts,
    ExportResponse,
    GetDocumentationPartsRequest,
    Integration,
    IntegrationResponse,
    IntegrationType,
    ListOfPatchOperation,
    ListOfString,
    MapOfStringToBoolean,
    MapOfStringToString,