Yes, you delete the original and enter the corrected 1099. You need to pay any additional tax by April 15th, but the 1040X does not necessarily need to be sent by April 15th.
The amended return (1040X) will show you the difference, so if it shows tax due, this will be in addition to what you paid with the original return.
It is best to wait until the original has processed, the IRS may even correct for the 1099, in which case they will notify you by mail and ask if you agree with the re-adjusted tax.
