Can I sell the shares on the record date and still get the rights issue benefit?

Yes, you will be eligible for the rights issue even if you sell the shares on the record date. If you sell the shares on the record date, you would still own the shares of the company in your Demat account as on record date as these will be debited from your account post the record date.

To know more about Rights Issue, get the SEBI FAQs here, NSE FAQs here and BSE FAQs here.