Your rationality scares me because you are taking your emotions out of the equation, i understand that is because they are too painful for you just now.
"Why would i go to the doctor"? You go to the doctor because that is how you will access help. Everyone is different but one thing they will NOT do is touch you, you are going there with a mental health problem, there is absolutely no need for a physical examination, it just wont happen, i promise. If you are worried about that, you can ask for a nurse to be present. I am not an expert, but i have been where you are now - had the siimilar, if not the same thoughts, both after DD was born when i had PND and recently when i had a breakdown - this last time was much worse. I would make these reasons in my head why my DDs would be better without me.
I was where you are - but the difference was I knew i was in danger and i didn't feel safe so i went to the doctors. I told my doctor, who i get on with well to be fair, what i was feeling and that i needed help. I was referred to the CMHT and seen the following week by a counsellor/psychiatrist for an assesment. I had already started on anti-depressants and was waiting for them to start working. There were plenty of questions yes, but nothing that required "the right answer" there isn't one, but these people are trained and can glean alot from those conversations. I managed with the help of my DP to get to a place in my head when i felt safe and was put on the list for counselling - the lady from the mental health place phoned me every couple of days to see how i was, i was given instant access to my GP and telephone numbers to call if i felt unsafe. There was no drama, no question of admission into hospital (i woldnt have wanted this, like you, the thought of being away from my DD is too much for me) it was all very calm, friendly even - the most iimportant thing - the ADs did their job and I have stopped counselling now - iim not 100% but i know i am not going to one day walk in front of a train. I take pleasure in little things - a nice cup of tea, a glass (or three) of wine - spending time with my DD.
How old are your children? Please don't beat yourself up about not being super crafty mum - WTAF is that all about anyway, the need to produce all these wonderful creations? sod that for a game - that is why nursery was invented, i try with the craft thing, im no good at it - my dd is right now sat at the table decorating biscuits - from a packet that i brought from the co-op - she's asked me to help but im actually quite stressed at the moment so she is having to do it herself, shes doing ok - im not a perfect parent by a very long way, i shout, i get stressed over the tiniest of things and im a bit shit when it comes to playing with DD. She is 8 now - she tells me im the best mum in the world, i tell her "well, im the only one you have!" And thats it really isn't it - im a bit rubbish at this parenting lark but my DD doesn't see that - i am her mum, i am her reality, she guages it on me. Yeah other mums do baking and drawing and such like - i am just not that mum - and that is ok, i have come to realise that mum is trying too hard!
What support do you have wiht the children just now? are you getting them out to playgroup? are they at school? Do you have a surestart or children's centre near you? If you go to your GP, one of the things they can do is point you in the direction of support - i used to go to homestart playgroup, it was my lifeline when DD was young - it was brilliant, i'd walk in, one of the helpers would play with DD and someone would put a cup of tea in my hand, sometimes id talk to other mums, other times i'd just sit quietly and watch the children play. Once a week i'd do this, it made such a difference. I didn't know it was available, i came across it by accident, if i'd asked at the docs i would have known before.
I hope you are feeling better - i don't know what has made you feel this way, but please know that it can get better. You are not well - if you had the flu, you wouldnt be questioning the use of going to the doctor and getting treatment - its much the same with our heads, only we are a bit more complicated.
Sorry for the epic post - been thinking about you all day x