Dear Tom Mangan,
Thanks for using Microsoft products and posting in the community.
I understand that you need to move your emails and contacts from Outlook to a new PC and this is causing you problems, I understand how you feel.
In order to better understand the situation of your problem, I need you to share some details of your problem with us:
1. What is the domain name of your email account?
2. What protocols are configured for your mailbox? (POP, IMAP and Exchange)
For more information on synchronization, see the following link: Sync basics - what you can and cannot sync - Microsoft Support

If you have configured Exchange protocol, then mail, contacts and other data will be automatically synchronized when you log in to your account on the new PC.
If you have configured IMAP and POP protocols, you need to export your emails, contacts using .pst file, for more information, please refer to the following link: Export emails, contacts, and calendar items to Outlook using a .pst file - Microsoft Support
You can make several backups as this file may get corrupted. Receive the .pst file in Outlook on the new PC by first logging into your account in Outlook on the new PC and then sending the .pst file to yourself via Outlook itself on the previous computer.
You can then use Outlook to open the data file (.pst) on the new PC, for more information see the following link: Import email, contacts, and calendar from an Outlook .pst file - Microsoft Support
Thank you for your patience and support.
Have a great day.
Best Regards,
Linda.L-MSFT | Microsoft Community Support Specialist