Trademark application rejected. Now what?

"Might we pass on appeal if we offer our vodka under a different name, but retain the name SixSpirits for the remaining types of spirits?"

An application for SixSpirits brand gin, or tequila, or another type of alcoholic beverage will likely face a similar, if not identical, rejection under DuPont factor analysis. If you decide to market your vodka under a different name, whether you may obtain a registration for that name will depend, of course, on what that name is.

Your example underscores the need to involve trademark counsel or, at a very minimum, thorough trademark clearance activity.

Normally, it is not advisable to discuss specifics in a public forum like this because you are not retaining any attorneys here and therefore have no attorney-client confidentiality. For all you know, an attorney or another affiliate of the company whose registration is cited against your application could be reviewing this website. But since you have identified your specific application, I'll mention a few pointers.
Your understanding of the office action illustrates your need to consult with a trademark attorney if you have any interest in proceeding with this mark. Trademark law is a lot more complicated than it seems and contains many traps for the unwary, including for attorneys not experienced with trademark law. Yes, the office action cites a registered mark against your applied-for mark because both marks contain the same dominant term, SIX, and both your goods and the registrant's goods involve alcoholic beverages. The other "thirty" or so "examples" are not registrations preventing your application; they are merely evidence the examiner presented showing how your alcoholic goods are related to the alcoholic beverage of the registrant. The point here is that marks can be considered infringing or too similar as long as consumers could confuse the two marks because of their similarity combined with the similarity or relatedness of the associated goods/services. The marks do not need to be identical and neither do the goods. Moreover, even if you could overcome this refusal by the examiner, the owner of the cited registration could oppose your application if they feel your mark would infringe upon theirs. The cited registration was renewed a couple years ago and the registered mark has been in use since 2001 so you have a serious issue with adopting this mark. This illustrates the importance of consulting with a trademark attorney to clear for availability any mark that you want to invest in prior to investing much in the mark.
Moreover, you mention that this application is for "A partnership [you are] involved with" which itself should involve a lawyer when discussing trademark rights. That is because a trademark is a business property, and an intangible one at that, which among partners often leads to a serious legal dispute when the partnership breaks up (which inevitably does). However, your application identifies your company as a limited liability company so I am not sure to what extent your company is structured like a traditional partnership.
This answer just scratches the surface of a number of issues you may be encountering not only with this mark but with any mark that you could pursue, especially if you try to adopt a mark on your own without any professional legal guidance.
*see the important disclaimer in my profile when considering this response.