I believe when you mention SingNet hosting, this is a commercial service, not referring to the consumer service.
Website access is via A record, Mail access is via MX record.
From the description above, your website is using bluehost, but both your singnet.sg and abc.sg domains email services are using SingNet email services. You will need to dump your emails from one account to another account. There could be some emails coming in while doing so, but that should be just a small number which can be manually copy over using an email client. You will then need to setup a forwarding rule in your singnet.sg email account to copy all all incoming mails to abs.sg email account.
Then for a year or 2, retain the singnet.sg email account until all your services or clients are aware of the new abc.sg email address.