Hi, I'm Sudipto. Welcome to JustAnswer. I'm reviewing your question now, and will post back with your reply ASAP
Unfortunately, you cannot update your new number, unless you are able to login to the account, so if you have forgotten the password ***** your account and the only recovery option is getting the code on the old number then, there is only one way left, which is creating a new account. If you wish then, I can help you to create one.
No cost for the google account, as it is free. But, yes for our service, as we are Justanswer.com a global paid support site and not google. Please let me know, what you want to do?