You will either have to get the old account working again to be able to sign out of it on the MacBook Air, or you will have to ask Apple to reset the Mac.
Apple has no "obsolete" category for accounts. My account is 20 years old and is not "obsolete". I don't know what you mean by "obsolete".
If you see a message below, or a similar "inactive ID" message
- "Inactive" Apple ID/Account
- "Verification Failed. This Apple ID is not active."
do as outlined in a post by Apple Community Specialist Brittany1416 in the thread: Inactive Apple ID - Apple Community
"We'd like to get you some help with this issue. For this issue, please contact Apple Support for assistance. You may be able to reach Apple Support here: Contact - Official Apple Support "
Apple can completely reset a device if a person can provide sufficient evidence of valid ownership. Check the requirements and steps at: Activation Lock support request - https://al-support.apple.com/#/additional-support Apple says, "Proof of ownership documentation is required," but does not detail what currently constitutes proof other than it “must include the product serial number, IMEI or MEID.” Apple may typically require the original receipt* issued by Apple or an Apple Authorized Reseller. If Apple resets the device this will give you access to the device but won't restore access to the account used to lock it. There is nothing that can be done to expedite an Activation Lock request.
*Apple recommends you "Keep your receipt after you buy an Apple device" --> Keep your receipt after you buy an Apple device - Apple Support