The required time for owning and living in the house, in order to avoid tax on the gain, is two years, not five years. You have to have owned and lived in the house for two out of the five years prior to the sale. That's where you saw 5 years. But you don't meet the two-year requirement.
Having one baby is not one of the exceptions. If you have twins (or more) you would be able to avoid tax on part or all of the gain, but the limit would be less than the normal $250,000 or $500,000 limit. If your gain is only $12,530 the entire gain might fall within the reduced limit, but that's only for a multiple birth, not just one baby.
Using the gain for the down payment on a new house does not avoid the tax. What you do with the money from the sale makes no difference.
