Additional contact-related features and issues within Microsoft Teams Free
What Login ID have you used when signing in to Teams application? Was this Login ID exactly the same as used with Skype?
Please, post a screenshot of the error message you received when starting Teams.
Also, login with your Skype account to Skype data export site and try to export the chat and file history.
https://secure.skype.com/en/data-export
Are you able to export these data?