Same problem with me two years ago. After much practice with timing, and balance training, and practice, practice, practice, I think I have pretty much overcome it.
Yes, I was getting water in nose, sinuses, and having serious reaction to chlorine for 24 hours after each swim.
Now, that reaction is pretty much non-existent as I continue to practice relaxed continuous exhalation under water. It just takes time and focus. I also usually do a very short, quick exhale just as my nose clears the water, to clear my nose prior to inhale -- sort of like a whale blows upon surfacing.