You could have a sinus infection in your upper sinuses. In many cases, it is a fungal infection that causes a constant allergic reaction. This is a recent discovery. Google "Dr. Jens Ponikau" to see his research on this subject.
Sinus rinses are useless in your situation, due to gravity. A Sinus Flush will overcome gravity. Use hypertonic saline, which will shrink the swollen tissues. You may also try nettle leaf capsules, which are a source of magnesium, which some studies have shown to lessen allergic reactions.
The Flip-Turn Sinus Flush is mildly risky, because you have to bend over to do it, preferably in a shower, but you can also do it outside on soft ground, or you can kneel down and lean over a bathtub with the shower running.
In my personal experience, before I invented the flush, I was unable to breathe through my nose at all, even when I didn't have an active sinus infection. My turbinates work well, now.
The feedback on turbinate reduction, at least on this forum, is less than favorable.