1. These do not sound like "normal ups and downs".
I'm just a unipolar depression sufferer and not a medical professional, so this is not in any way intended to be a diagnosis. It is instead meant to be a hint as to what others might see. These may be normal cycles for you with regard to your own life, but other people (who don't have the same tendency, whatever it genuinely is) might be very disturbed by what they see. That is a powerful disincentive to leaving you to yourself, even if you personally believe that you will be fine if you are left alone for a while.
2. Find out what they want to accomplish.
If the above is the case, then even if they understand that you sometimes go through darker moods they may not be interested in leaving you alone, even if that's what you would prefer. If this is the case it would be worth talking to them (your SO, for certain, maybe not your kids yet) to figure out their specific concerns and how you might be able to assure them that you are/will be OK. Questions like "You always seem concerned that you need to do something when I'm down. Do you know what causes you to feel that way, and is there anything I can do to reassure you that it's not necessary?" might help. The core of this is that it's not about you achieving your stated desire or your family understanding you better, but rather about you understanding them better. Maybe you can provide some reassurance, not necessarily in the moment, that helps them cope with your need to periodic solitude.
3. Consider developing a routine, if you don't already have one.
It may be worth designating a specific area as personal space that you use when in these moods. If you install yourself in a public area of your home and then demand that no one interact with you, you will be putting a lot more pressure on your family than if you retreat to your home office. A space or activity which helps to "normalize" the behavior in the context of everyday life may help a lot. This doesn't even need to be in your home. For example, you mentioned taking walks already. It also helps to give a signal to your family that you are in such a mood-- them acting normally and then abruptly encountering it makes it easier to believe that it's their fault.
4. You may need to give ground to maintain your relationships.
It's easy to think of this as something you personally need that your family is having trouble providing. But it's also the case that your family needs things from you, and your effectively disappearing for a day or two may not be feasible. No matter how important you feel it is to get this alone time, it might not be realistic to expect that you'll get 100% of that 100% of the times that you want it. That's not to say you should aim for 0% of what you want, but if you're viewing a single, brief interaction with a family member as "ruining" your alone time it may be easier (and better for your family) if you revise your expectations and satisfaction somewhat.
5. Seek a professional opinion.
I know you stated that you're not looking for help with your moods, but this is still for your family. I've heard depression described as seeing the person you love essentially die right in front of you, and good luck getting your family to shrug something like that off. As it stands it sounds like your family sees you suddenly and inexplicably descend into these moods, which distresses them. Any explanations and reassurances they can get will probably be valuable and help them feel OK about things. A professional opinion helps with that, and also is infinitely more valuable than your own subjective impression of "what you need".
Your description of these moods makes it sound like your ability to judge yourself and the world really changes, and consequently I would put very little faith that judgement as an outside observer. A psychiatrist or therapist saying that you need some alone time to move past the moods would be much, much better.