I'm not sure if this helps, but 'ovel' is a Hebrew word for to mourn. An 'ovelling dove' would be a mourning dove. I understand that mourning doves are very common in North America, and have a very, well, mournful call! The word ovelling also means moaning or lamenting (I looked up some old sermons online) which comes to the same thing.

Just an idea, but it could fit with the sentiments in the song...