If you and your partner want to move in together just go for it, I understand you are wary, but you've not rushed into anything, you deserve to be happy. Just pool your money together and I'm sure all will be fine. Your partner when he got with you knew about your son and I'm sure it's more of a pride issue on your part about him paying more. My daughter went through the same as you, in the end they went for it and once she swallowed her pride and pooled money together they were fine. Her ex never pays a penny been 4 years now, gone through CSA, but he keeps changing his job.
Good luck to you, when you have a chance of happiness grab it.