I would set aside the phone that’s not working and get control of your AppleID using a web browser on any convenient and secure computer.
By design it is difficult to bypass activation lock unless you are the original purchaser of the phone and can provide proof of purchase to Apple Support.
To work around this issue:
open https://appleid.apple.com, login with your old Apple ID and change the email to a new address (eg. firstname.lastname@example.org)
on the Mac/iPhone with the issue, add this new Apple Id as a secondary iCloud account
enroll with your Managed Apple ID
remove the email@example.com Apple ID from the device (which can now be deleted)
Is there a way to still get into iCloud to activate Find my iPhone?
Yes, you can login to iCloud to activate Find My iPhone. You can get a verification code via SMS or voice call on your trusted phone number.
Refer to this Apple Support document:
What if I can't access a trusted device or didn't receive a verification ...
Yes, in general you can request a password reset using the Apple private relay address. There is a small caveat in that the sender of the the password reset mail must authenticate their emails by registering their outbound email domains and using the Sender Policy Framework (SPF). This is a requirement for app developers / companies that wish to use the ...
I had an iMac showing twice within "Calls on other Devices" on an iPhone
The following got it fixed for me.
Removed my iMac from the iCloud account
Logged into iCloud again on the iMac
Logged in FaceTime on the iMac
Switched off/on "Calls on other Devices" on the iPhone
Hope this helps
Since you mention that you are running Mac OS X Lion (10.7.5), the issue could be that your operating system is no longer supported. Try to update your Mac and see if it works.
I had a similar issue on Mac OS X Mavericks (10.9). Once I updated to Mac OS X El Capitan (10.11), the issue went away.