Argh I just lost my post. I get exactly the same panic attacks for the same reasons. Whenever my DH goes out and I have to deal with my DCs by myself. This creates big problems, not only because of the panic attacks, but because I resent DH leaving me which causes problems between us.
In the short term, can you distract yourself from your anxiety by watching some comedy? I watch something like Father Ted or The IT Crowd - it distracts me enough that I stop thinking about it, and the fact that I smile and laugh makes me feel much much better. If I'm actually with the DCs and feeling overwhelmed, I will (after making sure they're fine - usually in front of the TV or drawing if I'm struggling) shut myself in a room, and either listen to music, go on mumsnet, or watch comedy clips on Youtube.
For some reason, watching comedy really helps relieve the panic.
In the long term, are you having any treatment for anxiety? I've tried a number of ADs for anxiety and depression - although they didn't touch my depression, both citalopram and sertraline massively reduced my anxiety. I also have oxazepam which I take when I start to feel panicky. Even the very fact that I have the oxazepam in my bag means I'm less likely to have a panic attack. If you haven't been to the doctor's, it might be a good place to start. It's not a sign of weakness to take a tablet to treat a disease. That's all that depression and anxiety are - a disease like diabetes that requires treatment and management.
Can you plan next week so that you have something to do every day? Whether it's go and visit a friend, have a friend come round to you, go to a new park you've never been to before etc. Sometimes we'd just end up going to a pet shop and looking at the fish - it was enough of a break during the day that I was able to break the day up into parts to get through it.
Also have you tried writing down the worst case scenario you can think of while you have a panic attack? Actually writing it down can help fix down the worry or the fear so you don't have to revisit it again and again.
I hope some of it helps - I have every sympathy, I really do. I work full time because I find it much easier than looking after my children alone.