Yeah... I find I tend to shut down as well when faced with big problems..
When I'm shut down, I just want to be alone, or do things that have no relation whatsoever to the problem, and become tied up in the activities.. Then I tend to do that all day, and then something comes back to bring me back to what I was supposed to deal with and it's even worse for me. So i know it's just an escape and it won't change the problem.. But I get stuck in the mind about it..
I found that if I don't get so attached to my thoughts and like you say, just do things one step at the time, trying to really stay detached from everything then I can handle things much better. But it's getting to the detachment that is hard..
The way I stay detached though is key, and I do that by grounding myself in the moment. You know, like they all take about awareness, the "now", being in the present, etc.. but it really works, well at least for me, it really does.
They give techniques about how to do it, but the way I do it, is by bringing myself to where I am, and using my senses to ground myself.. But again, I really try to stay detached from the senses, or otherwise, I become fascinated. So I guess the more I did it, the more I got better at it.
So when I get into it this shutting down, I bring myself out by trying to feel things with my hands and focusing on the feeling, then on hearing, and then on seeing.. But I try to do this in a quiet area first, so that I can handle things without being overwhelmed. That's when I can feel a lot more relaxed, and then I try to look at the problem without identifying it to me. Just looking at it for what it is, and what needs to be done. And like you say, little by little I see what needs to get done first.. Then I can make a list, and do things one at a time. In bad cases, I just need to do this over and over.
it's been several years that I've been doing that, and it's really helped me to overcome a lot of anxiety, stress and shutting down behavior..
Good Luck