I don't have a solution but I am in the same boat. I can't sleep at night and am constantly hit. I know it's summer but I keep my house at 69 70 all night long. I'm 12 weeks too.
I would say sleep when you all can. I had a hard time sleeping for 3 weeks straight and It caused me to have lots of anxiety attacks and put a lot of stress on my baby which caused me to be admitted into the hospital.