They are inconsistent with the volume numbers on issues of The Morning Star. When Hazel first arrives in New York, the front page says it's issue is in Volume 27. Several days later, when Hazel blacks out from drinking too much, it's listed as being in Volume 22 (which would be roughly five years earlier in most real world publications).
From inside the plane they were flying in, Wally Cook (Fredric March) and Hazel Flagg (Carole Lombard) would have been able to read the sky-written message stating 'Hello Hazel' welcoming her to New York, but it would have been viewed backwards to anyone on the ground below, the presumed audience for it.
In the final scenes aboard ship, the sun is setting over the waves in the background. The waves are breaking, however, indicating the background was filmed from a beach rather than from a ship out at sea.