Can my fiancé claim my son and I as well as our new baby if we are not married?

The new baby can be claimed by your fiance as a qualifying child if the birth was December 31st or earlier for 2017. 

Your son, if not the child of your fiance may be a qualifying relative, but not a qualifying child because the relationship rule will not be met until you are married. He cannot be the qualifying child of another person such as his Father.

You might qualify as a Qualifying Relative if you had income below $4,050 and you lived with your fiance all year. Click the screenshot attached to enlarge and view.

The rules for a qualifying child are show here. 

• Your son, daughter, adopted child1, stepchild, foster child2 or a descendent of any of them such as your grandchild
• Brother, sister, half brother, half sister, step brother, step sister or a descendant of any of them such as a niece or nephew

• At the end of the filing year, your child was younger than you (or your spouse if you file a joint return) and younger than 19
• At the end of the filing year, your child was younger than you (or your spouse if you file a joint return) younger than 24 and a full-time student
• At the end of the filing year, your child was any age and permanently and totally disabled3

• Child must live with you (or your spouse if you file a joint return) in the United States4 for more than half of the year

• The child cannot file a joint return for the tax year unless the child and the child's spouse did not have a separate filing requirement and filed the joint return only to claim a refund.