Any good bird hunter including me will tell you that you need both. I used to run a 12, 20, 16 and a 28. I shoot doves one night with my 12 and the next night with my 20 or 28. I guess I like the diversity and changing things up. This fall I carried both my 12 and 20 for grouse. Next week on the opener of South Dakota pheasants I'll hunt the first couple of days with my 12 and then probably switch later in the week to my 20. When I go back to South Dakota for a 2nd hunt in November the birds will be wilder and I'll stick to just my 12.
If I was you I'd start by finding out what you like to shoot an overunder is my preference because i enjoy shooting them and shoot them fairly well. I even take my overunder in the duck blind and goose pit which most guys I hunt with think is crazy. I just don't shoot pumps or automatics anymore but I used to when I first started. I just enjoy the feel of a well balanced gun and for whatever reason I usually find that in stacked barrels.
A slug in a 12 or 20 will stop a bear or any other thing walking the planet. I grew up in Iowa and took many deer as a kid with both a 12 and 20 slug they do an incredible amount of damage. I was gifted a Mossburg 500 years ago it's versatile but to me swung like a fence post taking a lot of enjoyment out of wingshooting. I think there are better options out there than Mossburg you should take a look at. Good luck, bird hunting has been an addiction of mine for over 50 years hope you enjoy it as well.