Solow, I have only just found this thread, but I too could have written your post 9 years ago when ds was born.
I just wanted to second everything that everone else has already said and to offer you my support too.
The time of year is not helping you, I'm sure. Its no fun having to get up (or not sleep at all) when it is freezing cold and miserable outside. But, the evenings are already getting lighter and with every week that passes, another quarter of an hour of daylight. Before you know it, it will be british summertime again and you will feel so much better. Flowers will be sprouting and then flowering - look out for them, it helps.
The other thing I wanted to reiterate which has already been mentioned is to get out and go to toddler groups if at all possible. I didn't realise you were "allowed" to go with a baby with my first, but you will be welcomed and given so much support by other mums.
Both my children were like your baby, but it was so much easier with dd because I knew it wouldn't last long.
Well done for posting here, come back and keep us updated and enjoy your little one. Remember, a baby that needs constant cuddles is Nature's way of making you rest too. So, forget the housework and enjoy these precious times.
So quiet down cobwebs, dust go to sleep; I'm rocking my baby and babies don't keep. I love this poem x