Is there another StackExchange Site that we could redirect Intrapersonal Questions to?
Short answer - no.
For the academic side of psychology and cognitive science, there is a Psychology SE, but they explicitly disallow self-help questions. Self-help questions are also off-topic at Lifehacks SE, and Productivity SE, which I suspect is the closest to a self-help stack thus far, was shut down for lack of activity...
The Psychology SE meta post has a great explanation of why self-help in general doesn’t work well with StackExchange:
Stack Exchange in general discourages questions that are "too localized", meaning that they only likely apply to a small set of people, and will not be very useful to future visitors looking for material on that particular topic. Since everyone is quite different, it's difficult, if not impossible, to generalize advice, and even if we could, we shouldn't.
Can't we just try our best and dispense what we think will work? Well, why shouldn't we, this is the internet and people come to Google first for help all of the time, right? Certainly, I can fix my own brake hydraulics in my car using instructions from Joe's Auto-matic.com (sorry Joe), but I'd hate to put my own safety at risk by not taking the advice and harnessing the experience of a professional who deals with this type of thing all of the time, and is certified to do so.
Likewise, in cases where you are seeking help, you should seek advice from a doctor or licensed mental health professional who can see you in person, learn the important parts of your history (medical, social, and psychological, without you having to share it with the interwebs), and can tailor a treatment or other plan that is right for you. How can we tell if your "brief funk" isn't due to an autoimmune disease, and someone telling you "just smile your way out of it and read Aaron Beck's books at bedtime" isn't going to put you in danger? We're thinking of your own safety here, and not in a patronizing way, but in an "it's your own safety" kind of way!
But, you ask, what if the person answering the question is a licensed doctor, psychologist, therapist, or nurse practitioner? Well, laws vary in terms of right to practice under certain conditions, and any quality clinician should want to see you in person, at least once. I think you would find that most responsible clinicians, even if they could treat over the internet, wouldn't risk their license to provide you with sub-optimal care anyway.
Another point that is kind of implied by all that: StackExchange is intended to curate expert Q&A. We can expect to find some experts in conflict-aversion... but there's really no such thing as an expert in user123s-inner-self, aside from User123 themself and maybe their therapist or very close friends.
Finally, it's okay if there's no place on the network for these questions. It doesn't mean those questions are bad -- it just means that they don't fit in this particular box. There are many other communities on the internet willing to take on topics like self-help, relationship advice, how to cope with depression/anxiety, etc., and OP is totally free to search those up and post there instead. (I participate in a couple such forums myself!) An actual discussion forum will also allow you to carry on conversations, ask many follow-up questions, and share stories in a way that StackExchange discourages.