Xbox 360, playing it without signing up to Live. Plenty of achievements. I wanted to sign...

Hello Roger, sorry to hear you're having trouble with your account.

You should be able to create a new Hotmail/Outlook email address (via the email website) and then use that email for the account on the 360. When you're logged into the offline account select the option to join Live and then use the new email that you've created and add it to the offline account when prompted.

I had the exact same problem. No achievements showed up on my Xbox One after "Animal Cruelty", but they did if I used the achievement section in the Xbox 360 guide. So I tried some of the same things you did and even started a new game to unlock "Acceptance",
and see if that would unlock properly. It didn't. Then I noticed that in the 360 guide I wasn't signed into Xbox Live, even though every time I played a notification popped up that said I was "signed in", it apparently didn't mean signed into Live. So I clicked
on the button to sign in, got a error code, and based on that reset my router and modem. Went back into the game and signed in to Xbox Live through the 360 guide and that solved the problem. I hope this helps.

It is possible of achieving what you want with one Gold sub, but it would require one console to play on Live as your account.

You set up Console A as your Home Console, then any account on it can use your Gold sub without your Gold account being signed in, then on Console B you sign in with your Gold account to access Live.

Sadly whenever you play that particular game, it's going to sign you out of Xbox Live and you'd basically be playing an offline version of the Xbox 360. No connecting to Live in-game or achievements.

You will keep your achievements and game saves when Disconnected From Xbox LIVE and you can continue to do those things when not connected. Play games, get achievements, save games to your harddrive etc.

You are on a Free Xbox LIVE Account and it will remain like that forever now with seeing Dashboard Advertisements and other tabs and getting Updates. You cannot revert to an account before the Xbox LIVE Sign Up Process, unless you create another Gamertag and
do not Sign Up To Xbox LIVE or simply remain Disconnected from your Internet Provider in the Settings tab I explained.

What is it that you do not want to about being an Xbox LIVE Free Membership? There is no fee and everything is how it is before you signed up to Xbox LIVE. You are not paying for anything and never will (unless downloadable content etc which is OPTIONAL) when
signed into an Xbox LIVE Gamertag.