You need a plan. Firstly, how likely is it that your ex will ask you to leave? What discussions have you had about future living arrangements and timeframes? If you haven't yet had this discussion then this would be a good place to start.
What qualification are you studying for, how long do you have left to complete the course and how does this affect your future job prospects? These things need to be taken into consideration before deciding whether or not to quit your course.
You say it his his house, does he pay a mortgage, how long have you lived there and have you contributed towards the mortgage at all? You need to establish if you have a beneficial interest in the house, just because you are not named on the mortgage doesn't mean you don't have a beneficial interest. It may not be much, but it may be enough for a deposit. You need to take professional advice on this.
What is the plan for future childcare arrangements? Will it be 50/50? Again something to discuss with your ex and make a plan for the future.
I was in your position about 18 months ago. We have been separated but living together for 2 years, but we had a plan, and i will be completing on my new house in January. If you have a plan and a clear direction and timeframe then this makes things easier, and it has got easier as the time has gone on.