I am very sorry for the issues.
Unlike Yahoo! and AOL (any many other e-mail providers), Google / GMail does not have phone support nor any other human contact available. For that reason, you cannot call them and explain what happened. The best you can get from them is an automated response to the Account Recovery form telling you that you either provided sufficient information for password ***** or account verification, or that you provided insufficient information, or in case, that you need to use a different computer.
The only other method I can suggest you is to call your Internet Service Provider and ask them if they could assign the same IP address to your new modem that you used when you had the old modem. That will allow you to be recognized as the owner in the GMail system, so you shouldn't see the verification prompt. Once signed in, you can change the recovery e-mail address.
In case you still have the old modem, connect it and check again.
If nothing helps, the GMail account won't be accessible to anyone and will automatically be cancelled (expire) after 9 months.
If you plan to open a new e-mail account, my recommendation is to open a Yahoo! or AOL e-mail account since they have a phone support you can call in cases like this one.