Hello Littleray - congratulations on the birth of your third DC! Your post really struck a chord with me because two years ago my third DC, a boy, was born. My DDs were 3 and 1-and-a-half at the time and I remember having exactly the same feeding problems you describe.
Looking back, the reason the breastfeeding was so hard was because - as I think you have already mentioned - I was constantly on the go. Taking DDs to pre-school, appointments, playdates, shopping trips. The house still needed to be run, food made, washing washed, bills paid, shopping done etc. Poor DC3 just had to fit in, and that meant giving him a feed when I had the time, as opposed to when he asked for it. Often I would rush feeds because I had to 'be somewhere'.... My milk, I think, was also quite 'weak' because I was rushing around and not resting at all.
I ended up getting so stressed that DH and my Mum gently suggested the idea of switching some of the breast feeds to bottle feeds and trying combined feeding. So I gave him a bottle during the mid-morning and mid-afternoon feeds and this really helped. I found he would drink the bottle more quickly, freeing up a bit of time, and other people could feed him too.... which freed up even more time!!
Having three DCs under four is bloody hard work! When I look back at that period (which wasn't that long ago) it was a complete blur. All I can remember was DH and I constantly just toiling away at the coalface. It was relentless.... dealing with feeds, nappies, toddler meals, toddler tantrums... it was a slog. DC3 was also terrible at night due to wind/colic or some such problem, so DH and I were snapping at each other permanently through lack of sleep.
Is there any way you can get any help from friends/family? Can you recruit any grandparents or aunties and uncles to help take some of the load off you so you can rest? It's really important you find time for yourself as you are the vital cog in the wheel that is your family and if you crash and burn, it will all cave in! So, don't feel bad about taking an hour or two just ... well... sleep! Or even to just read a magazine, or to drink a pint of water, or make yourself something healthy to eat. I was crap at asking for help, and that was my downfall. Some of my friends who did not have family living nearby used to get the teachers from the nursery to help out and come and sit with the DCs while they went and had a rest. Little things like that can really, really help. Could you ask the nursery teachers for help?
Sorry for the long post. Please feel free to PM me if you need any more advice - as I have totally been there - and come out the other side totally unscathed (although maybe a few grey hairs)! Good luck!!