Oh my...I don't suggest actually breathing air into her, as you can easily rupture her Abdominal Air-Sac doing that...Bird's lungs don't actually fill with air or expand, instead all the air they breathe-in fills their Abdominal Air-Sac, and if you use too much pressure it can rupture, and then that's it...
I don't quite understand how the medication is going up into her sinuses, as it should just go right down into her crop...At most you're giving her 1 or 2 drops of medicine, so I think maybe you're actually sticking the oral syringe way too far into her mouth/beak and that's why this is happening. You aren't actually trying to put the medication directly into her crop, as you will never be able to do that with an oral syringe; the only way you can ever put anything directly into the crop is with a "Crop-Needle", which is a very long tube that you actually insert right down the entrance to the crop and then you check to see if you can see the end of the crop-needle moving inside of her crop by looking at her chest, and then you inject whatever you're giving them into the crop directly...With an oral syringe, if you are actually trying to get the tip to the back of her throat where the opening to her crop is, you're probably causing her to actually breath the medicine in, which is very bad...The last thing you want is to aspirate anything into her lungs and cause Pneumonia...
All you need to do when giving her the drops of medications is to hold her in one hand and make it so she is facing you, then you insert just the tip of the oral syringe into the left side of her beak (HER left side, so your right side when you're facing her), and then aim the tip of the syringe over her tongue and across her mouth at a diagonal towards the back of the right side of her throat (HER right side, so towards your left side when she's facing you), and then you just inject the drop of medication into her mouth, not directly into her crop...The idea being that you're putting the medication, hand-feeding formula, etc., whatever, into her mouth but on the side of her mouth where the crop entrance is, and on top of her tongue, so that when you remove the tip of the syringe from her mouth she's going to automatically swallow it and it will go right into her crop instead of into her trachea...I think you're actually trying to force it into her crop and it's freaking her out, or you have too much of the syringe in her mouth or are using too much force and she's having trouble breathing, and she inhaling the medication right into the sinus opening at the back of her throat and right up her nose, just like we do when we inhale something we're eating/drinking and it comes out our noses...
The other possibility is that because one entire nostril is completely blocked that she's actually been breathing through her mouth all this time because it's too difficult to breath through her nostrils (too much resistance due to the blocked nostril), so since she's breathing through her mouth in the first place, when you are putting the drops in she gets nervous/upset and her naturally her breathing increases, and then she's breathing the meds right up into her sinuses...
Either way, make sure that you're only inserting the very tip of the syringe into her mouth/beak, ALWAYS ON TOP OF HER TONGUE, and just quickly push the drop of medicine into her mouth and then remove the syringe from her mouth, so that she actually swallows the medicine instead of breathes it up into her sinuses...
***This total blockage of one nasal passage needs to be resolved quickly, because it may very well be causing her to breath through her mouth and be causing her much stress and distress that is making her worse. The blockage is most-likely just a bunch of mucous from both her upper respiratory infections and also a bunch of mucous/pus/gunk from the conjunctivitis in her eyes, as they directly drain right into her sinuses...All of that junk has probably collected in that nasal passage, luckily far enough along that it's only on one side and not both...
***However, the other possibility is that the nasal blockage came first and is the entire cause of the infection to begin with...The only way to know that is to take an x-ray to see if the blockage is fluid/mucous/pus or if it's a growth of some kind, like a Polyp or a Cyst...An regular x-ray will show this immediately, but so would a nasal flush, because if the nasal flush with sterile saline doesn't clear the blockage and the flush only comes out of one nostril and nothing at all comes out of the blocked side, then you know she has a growth/mass causing the blockage, and it's probably been there for quite a long time and was actually the cause of all of this mess...
If you want to attempt a nasal flush it's not difficult to do, but you do need to have sterile saline available, along with a clean syringe with a small tip on it, and a syringe that will hold at least 6ml-10ml of the saline....It's not difficult at all to do, you simply hold her over a sink, with the blocked nostril facing down and the open side up, and then you simply fill the syringe with either 6ml or 10 ml of the sterile saline, whatever capacity syringe you have access to, and you put the tip of the syringe right up against/into the nostril on the open side that is NOT blocked (should be facing up), and then with gentle but even pressure you push the sterile saline into the unblocked nostril, and continue to push it in slowly but steadily, without stopping, until the saline either starts running out of the blocked nostril and you continue to flush all the saline in the syringe through it, or until the saline starts running back out of the side you're putting it in...If that happens, then it's a sure sign that it's a mass/growth blocking the nostril and not exudate...