Wow, you have a lot on your plate, well done for even etting dressed in the morning, especially if you dh is avay so much.
When I read your post, it could have been written by me a few years ago, I was in a very similar place!
Firstly, ds1 had terrible reflux as a baby, he was on all kinds of meds, but what actually cured the reflux was a cranial osteopath, so I'd really recommend trying this. That was 10 years ago, and my gp thought I was crazy, but now I think it's very widely recognised as a successful threatment for reflux. See if you can find someone locally who works with babies. If you can get some sleep you'll feel much better about yourself, and life in general.
Secondly, I hate to sound like my mother when I say this, but you need to eat to be able to keep up with it all, particularly as a vegetarian. When my first two were little I had a tendency to live on toast and lucozade, so no wonder I felt terrible. Try a bowl of porridge for breakfast, I do it in the microvave with a handful of frozen raspberries, and milk, and it's supprisingly good, and really keeps me going all morning. Milk is great for protien, allow yourelf a hot chocolate mid morning or afternoon, and I find a great thing to have in the cupboard when I don't have the time or energy to eat is a can of protien power milkshake stuff. You get it in the health food shop, comes in different flavours,mix with milk. It's not a gourmet meal, but it's agreat source of protien which will help you keep going.
Totally agree with you on having a good coat, it's an easy way to look smart without time or effort, no-one knows what's underneath. To be honest in this weather the warmest coat is the best option, but you could liven it up with a nice hat and scarf, have a look in Accessorize.
When spring comes, I think a trench coat looks great, belted to emphasise your waist. With jeans and converse, you'll look really smart.
People on here always recommend Laura Mercier tinted moisturiser, or if you are near a Bobbi Brown counter ask them to do a make over for you, her products are great, and I lke her, she always seems to be on the side of women who don't have enough hours in the day IYSWIM.
Hope some of this is useful to you