If you have server access you should be able to roll back to an earlier version on your site. Most probably it would be the php that changed and both G2 and the unmodified G3 wont run on that, G3.1 will run on php7 and uses sqli for the db
If you don't have access then ask the Techs at GoDaddy if they will do it for you so that you have a working G2. They won't like it but tell them you need it for a short while so that you can upgrade.
Once you have it working make a backup of your sql db and the full size photos then setup a neg G2 on the other site and upload the db and photos to it. Once it is all working put a permanent redirect from the GoDaddy site to the new one, that way everyone's links will update to the new site.
Then think seriously about changing to the new G3.1 as all hosts that have not done so will upgrade the php and you will be locked out.
I believe that you can do a server to server transfer but am not up on that method.
Others may chime in with a way to back it up even while broken but I don't know if that is possible.
Tony