Yes, you can appeal an unfavorable office action decision even if the USPTO issues a final refusal and your response hasn’t resolved the examiner's initial objections. In cases like these, you would typically appeal the decision to the Trademark Trial and Appeal Board. It’s an administrative tribunal that reviews decisions in trademark examinations.
An appeal must be filed within three months of the final refusal date. If the TTAB also rules against the applicant, the decision can be further appealed to the US Court of Appeals for the Federal Circuit or to a US district court.
The US isn’t the only jurisdiction where appeals are possible; other national IP offices have similar systems in place. For example, the EUIPO has a Board of Appeal, the UKIPO has an Appointed Person, and similar bodies exist in other offices. The process and deadlines vary by jurisdiction, but there’s always a system in place that allows you to revisit refusals.