Do I have to resign in order to claim constructive dismissal?
Yes, you do. You must have terminated the contract by resigning.
The law allows you to give your contractual or statutory notice when you resign and still claim constructive dismissal. However, you should give no more than the minimum notice required under your contract, Otherwise, your employer is likely to argue that the relationship has not irreparably broken down after all.
When you resign, you should spell out in your resignation letter that you are leaving your job because of the employer’s fundamental breach of the employment contract.