OK good on no blood or mucus, and good that you feel it sounds less like ILT!
Did she come with this, or did it develop at home? It could be that she's been introduced to the things your poultry carry (even through your own shoes, clothes, hands, pens, etc even if seperate) and is fighting a respiratory illness (viral or bacterial) and will overcome it. There are other things that can also cause gaping like old bouts with fungi in her respiratory system, dust, gapeworm, etc. The weepy eyes and shaking head could be one thing or many. And the nature of bacterial diseases is that you want to try to avoid treating with medicine X if medicine X isn't the right one for bacteria A.
In an ideal world, she would be taken to a vet, a culture and sensitivity would be done, and she's receive the exact antibiotic based on the exact bacteria grown and identified from her sputum.
But this doesn't often happen. In the mean time, she should receive supportive care as any ill bird would - particularly those with respiratory issues.
First, vitamin A is essential for healing the mucus membranes and respiratory tract. It is an oil vitamin and is sensitive to light. For that reason, I recommend that sick birds receive it on their food or directly in their beak - not in water preparations. Some studies found certain oil vitamins to be up to 45% more effective given in conjunection with food versus with drink. . The most wonderful solution I've found to that issue is simply Enfamil baby vitamins, Polyvisol (not the added iron formula). I found ours at our local
Walmart - vitamin section - infant vitamins. Three drops in her beak daily should help. Some of the symptoms of vitamin A deficiency are water eyes, excess mucus in the throat, and sometimes nasal drainage. Vitamin A is the vitamin that most often degrades from feed because of its sensitivity to light and heat, and because it does better in oil (not dry) conditions.
If you give the polyvisol, you will also be giving other helpful oil vitamins - like E which is wonderful as an antioxidant.
The tear ducts open connect with an opening in the roof of the beak (the choanal opening) which in turn connects with the digestive tract. Drainage and possibly infective bacteria change the pH and beneficial bacterial lode of the digestive tract. So it's not uncommon for birds with drainage to get secondary diarrhea. To prevent this, I would give her a probiotic daily through this illness and then every other day for 2 weeks after her symptoms abate.
Unless you decide to medicate with medicines containing an active ingredient ending with -mycin or -cycline, you could use plain unflavored yogurt at 1 teaspoon per day. Some birds love it, other birds need tricking. Another option would be to pick up acidophilis capsules from the vitamin section while you're getting the polyvisol. The capsules or tablets are easy to use as the powder you get from emptying the capsule or crushing the tablet is easily mixed with a tiny bit of water, or anything really, and given to the bird via beak. If the bird loves bread, one capsule sprinkled onto bread might not be noticed. Or egg yolk.
You want the good bacteria to thrive and be strong and the living bacteria in those products helps to assure that. It will also help her absorb more from her foods. She needs full nutrition for full healing.
If you feel she has mucus in her throat (indicated by her gaping), using organic apple cider vinegar (1 teaspoon per gallon of water) will help cut through the mucus. It additionally also contains living beneficial bacteria, and can even help readjust the pH of the gut to be more friendly to your GOOD bacteria army, and unfriendly to the bad bacteria - including those that slough down from the respiratory system. Only use organic as it's the one with the "mother" (living bacteria) and other nutrients.
These treatments would be necessary even if being treated with an antibiotic, but starting them now might facilitate her healing so that she recovers from this on her own.
Of course, she needs to be on the best food you can offer. At least 95% of a complete age-appropriate ration. The rest can be treats, grains, etc.
If you have questions, please feel free to email me (as my PM box is usually full) or contact me through this thread.
I'll help in any way possible.