Hacked email sending emails to my contacts then infiltrating their emails

You can create a new email account and redirect incoming messages from your old address, but you cannot permanently close the compromised Outlook.com account and continue receiving forwarded emails from it. Microsoft requires the original mailbox to remain active for forwarding to work. If you still have access to your old account, I recommend securing it first: scan your device for malware, change your password, review your security information and recent sign-in activity at https://account.live.com/Activity, and use Sign out everywhere under https://account.microsoft.com/security. This can take up to 24 hours. Also check Outlook.com under Settings > Mail > Rules and Forwarding and IMAP to remove any unauthorized rules or forwarding addresses. Once the account is secure, create a separate Microsoft account with your new email address. Then, in the old Outlook.com mailbox, go to Settings > Mail > Rules > Add new rule, choose the specific senders whose emails you want to keep receiving, select Redirect to, enter your new email address, and save.

However, keeping the compromised mailbox active means you must first ensure that the unauthorized access has stopped. If you permanently close your Outlook.com account, forwarding will stop, and closing the associated Microsoft account also affects services such as OneDrive and other Microsoft products. You can review Microsoft's account closure instructions here: https://support.microsoft.com/en-us/accounts-billing/manage/close-your-outlook-com-account.

To better determine why those messages keep being sent, could you confirm whether the repeated emails appear in your Outlook.com Sent Items folder when you access https://outlook.live.com/mail/ directly, and which security steps you have already tried?