You are absolutely right, you'll need to get a new SIM and make a new account with a new username.
Once you activate the new SIM on the new account, you'll also get a new number.
So, you would have 2 mobile number - 2 accounts and 2 usernames.
You can keep the payment details/names and email address the same for both the accounts though.
To get rewards, order a SIM from your old account (active SIM) https://www.giffgaff.com/orders/mgm
It'll get there in 3-4 days, activate it on the new account and top it up.