Hi,
The system takes into account the "Override shipping address" setting of the selected shipping method before assessing if it needs to check for the shipping method validity against the custom address fields.
So, normally, an error should not happen in such case.
Also, supposing this would happen, the error message should read "The shipping address you selected cannot be used as some information is missing. Please edit it before proceeding." and not "The billings address you selected cannot be used as some information is missing. Please edit it before proceeding."
So, it seems to me that your issue is with the billing address not being complete in your user account after you added some extra required custom address fields. Please check.