It depends on the cancellation policy with the first one. If it's on a strict or moderate cancellation policy then you're not guaranteed a full or even a partial refund if you cancel mid-stay.
With flexible cancellation you would be refunded at least 17 out of the 19 nights.
When Airbnb says they'll try to help what they mean is they'll contact the host nicely and ask if they are okay with giving a full refund, but ultimately it depends on the host whether he/she is willing to give you the refund in this case.