Then after you are sure you will NEVER need it again, you can delete for yourself from the primary billing account, select My Profile >> Manage accounts tab. This only works for secondary addresses. If it is your primary address it can't be completely deleted, but VM say that after 150 days of non-use, the mailbox will close.
I recommend backing up any emails you want, then deleting all emails and contacts from the VM server via webmail first. After the deletion process, double-check the mailbox is gone from the VM servers by running the address throughhttps://mailtester.com. Then you can be sure it is completely finished.
Do not use the holiday/out of office type notice that replies to everybody- you don't want spammers and scammers to get any information on your new address. But yes, the autoforward is safe so you can pick up on emails you need and can then notify the sender or ignore if you don't want them to know your new address.
Turn off all filters and blocklist. Turn off the spam filtering so you get any mis-identified spam forwarded to your new address.