Yes you can each claim one child. You can claim your daughter, and he can claim the baby. If your fiancé is not the biological father of your daughter he cannot claim her. Even if he is, she has been with him for less than 6 months so he couldn't get the same benefits that you can for claiming her. But he can claim the baby because the baby was with both of you the rest of the year, and in TurboTax he can say his biological baby was living with him all year.
Can you both claim Head of Household? Probably not. This FAQ provides more information: https://ttlc.intuit.com/replies/3288637.
Remember that both you and your fiancé are claiming the same address at the end of the year. Supporting a Household is considered over the course of the year, and the IRS' begins with the premise that two adults living in the same home cannot both be Heads of Household. The fact that you have children together would make it additionally difficult to make the claim also.
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