Christine, if you were single and left one of your jobs you'd get tax credits (the tax alone on the two jobs must be a nightmare!) which would give you top ups towards childcare, up to 70%. It would likely be possible to find a childminder who would be able to cover you for the early starts if you can't find a friend to give DD a lift to nursery. Or you could change your hours if it was the only option (even if it did cost more)
You would also be eligible for housing benefit which will be relative to the rental costs in your area, perhaps slightly below market value.
If you work around 20 hours a week at minimum wage, you will get top ups of almost all of this value and some on top, if you work 40 hours a week then it will be around half of this. Also, you would get some maintenance from your DP towards your DD and this is no longer counted against benefits.
Plus as others have said your life is very expensive just now because of his drinking - when you take this out you'd probably be surprised at how much you have spare. Maybe after bills you keep your money separate, but who buys food (is alcohol included on the weekly shop?), who buys clothes for DD, who buys cleaning stuff, toys, birthday presents, nice things for the home, pays for days out. Who pays for the car stuff (if you have one), who pays all of those other things that you can't set a direct debit for and forget? If he's spending the vast majority of his "spare" money on drink and you're tending to spend yours more on essentials or things which benefit the whole family, then that's not equal or fair.
I know the idea of relying on state handouts isn't nice, but when you're not slogging your guts out in two jobs at once and DD is a bit older and more independent and/or at school there is no reason why you couldn't get a more well paid job and bring you both out of the benefits trap. That's what they're there for - so you don't have to be dependent on someone who treats you like this.