This message is not intended as legal advice, and should not be construed as such. Speak to a lawyer or MOM.

That being said, legally speaking that's generally not true, moonlighting is only illegal in the civil sectors (to prevent conflicts of interest), otherwise, you are bound by contractual terms (no contract term preventing moonlighting in a second job means that you can).

The terms in the employment handbook are not enforceable, unless the employment contract mentioned that the other standard terms of employment are found in the handbook (and/or are incorporated into the contract). Furthermore, alongside the contractual incorporation, they also generally have to have provided you those terms (by providing you the handbook) prior to signing (or it was at least available upon request - whether you read it is another thing entirely).

You have a good case to seek the payment of your salary (if the contract does not ban moonlighting) and should consult MOM or free employment mediation agencies (TADM/ECT for example). Do not accept any offers of partial payment prior to seeking advice from MOM or a lawyer.