You could still probably blow with your mouth if you didn't have your lungs connected, I imagine it would involve a kind of burping type of action. I think the bigger problem would be that if your nostrils closed up, you wouldn't be able to breathe, and probably also talking would be a lot harder if your vocal chords and mouth were separate from your main air sacs.

I think the solution is probably just an easily opened and closed internal valve that separates the stomach and the lungs, rather than this bullshit we currently have with two separate valves that lead into both and open for one and then close for the other whenever it's required. It's still good to be able to close both when you want to, but you can already close your mouth on command, and another valve with the nose is a notable upgrade in that it keeps everyone from smelling bad smells they don't wanna smell, and it also doesn't take any more valves than we already have.

There's probably some way you could fix this all with enough surgical intervention, I bet...