I have this exact problem too! Having tried Beeper, and now using it every day, I can confirm it achieves this perfectly. I now just check Beeper in batches (t

I have this exact problem too! Having tried Beeper, and now using it every day, I can confirm it achieves this perfectly. I now just check Beeper in batches (there's a great shortcut for cycling through unreads), rather than having endless apps to check.