OK, let me enlighten you...I was a single mother...my ex husband and I moved in with each other before we got married. We also bought a house together before our wedding. HOWEVER, we paid cash for our house. We didn't have a mortgage.
So, it is ok to move in together. let him feel out what he is going to get (instant family)! It isn't easy. He is getting a package deal. Regardless of your childs (real) father! He will have to if he hasn't already, get used to the fact that your with this man. In my situation. My son's father was absentee! Then years later, popped back and gave my son a car, without running it by me first! (whole other issue!)
As far as buying a home before a your marriage. You and your fi, will have to qualify seperately for the mortgage. You will have to sign the loan docs alone. So, it is best to wait until after your married, so you can combine your incomes, and debt ratio to qualify for the loan. Improve your credit scores, etc.