Minecraft Account

You need to use the Mojang account's email address as your username instead of your original username.

The reason it may have still been working after migration is proberbly because your existing account credentials/session data.

It sounds like your best bet would be to use a VPN. It would allow you to be accessing a different network virtually and it should let you get around this issue. Or as you mentioned you can just play it solo offline. Networks often times at schools and/or workplace environments will have these network restrictions that basically blows network traffic or access to certain types of content, in this case it would be games.