I'm glad you found it helpful. I don't know you or your SO at all, so here are some general thoughts that help me:
> I have a hard time making the decision between listening mode and problem solving mode. Surely there are times when problem solving mode is helpful?
All I can tell you is that everyone seems to have this thought and the endless drive to insert themselves into the conversation, and we all find endless rationalizations and opportunities and exceptions - this time is special! - that will allow it. Me too. Recognize the drive; zip it. 'No, this time is not an exception - again'; just zip it.
It also means your attention, your thoughts, your emotion are focused on yourself, your ideas, your drive to talk, and not on your SO. That is a fundamental mistake of listening. Good listening takes your senses (visual, audial, maybe touch, etc.) to pick up what they are communicating; all your concentration (emotional on their feeling, intellectual on the words, etc.); composure (body language, etc.); emotion (they can see and sense it, of course); etc. Turn your attention away from yourself and focus it on them - like swiveling a big camera on a movie set, you are behind the camera, focused on the actors and silent now.
> I understand the "only when they ask for help" idea, but this specific person does not have the instinct to ask for help, which makes it difficult.
I don't know you or your SO at all, as I said, but here is a common trap: Maybe they don't ask because you already push it on them too often. They feel trapped, pressured; the last thing they want is more 'advice' - not that you have bad advice, but they need space, less pressure; they need to figure things out themselves without having your ideas fill the space, distracting their attention, and demanding responses. So they don't ask and then you see that and push more on them - a reinforcing cycle in the wrong direction.
Start a cycle in the right direction: If you back off, give them as much space as they want without resistance (or asking - 'can I tell you now?'), eventually they will flourish, come around, open up, and feel much happier and safer, and probably ask you. It may take lots of patience - and if you are struggling with the patience, you still have a problem with your personal drive and the focus of your attention (it took a long time for that to diminish in me).
> Or when they do ask for help, it is for such general large abstract problems that I am also at a loss of how to give actionable advice.
Look for the emotion behind their worry, and help them find their way through that. They are lost in the woods so help them find their way through; telling them what's on the other side doesn't really help.
I hope that helps!