From the DMV web site you gave
"Customers renewing Virginia driver's licenses or ID cards can keep your old license or ID card to carry with the temporary driving permit or receipt, until you receive your new license or ID card in the mail.
Your new driver's license or ID card will be produced within three days and mailed to you. Mail delivery times vary. Please allow up to 15 days to receive your new credential."
If I were you I would make sure to renew 2 or 3 weeks before your current liscense expires and you should be okay. Don't wait until the last minute.