I am sorry you are having so many eye problems. Those are one of the things I fear the most.
Is the eyelid swollen at all? Is there a sty on the inside (might look like a whitehead)?
Is the itching on the inside of the eyelid or is it more on the edge or just barely inside the eyelid? I get a condition called blepharitis that makes my eyelids hurt, itch, swell up (at one point my eyes looked like they had those Rolling Stone album cover lips around them!) and it is NOT fun. It is caused by the tear ducts and pores in the skin getting clogged by the skin not sloughing off the way it normally does. The dead skin cells plug the pores and tear ducts and everything swells and hurts and turns red.
If I get ANY irritation, I start using baby shampoo (NOT Johnsons - that no more tears line is a BIG LIE because it HURTS worse than any other kind of soap) to wash the edges of my eyelids and then I rinse very well with saline solution. You can buy cleansers designed to be used there, and they are awesome. When I was first diagnosis'd there really were not many of those and they were hard to find. I have a pump of the cleaner at home for when I need it and I have individual packets of premoistened pads designed to clean that area and help with this problems. They are always in my purse and they can be a lifesaver.
I still rinse with saline solution and then put in eyedrops. Anytime you use saline you should follow it with eyedrops because the saline can end up helping to dry things out, like licking your chapped lips makes them worse. Or so three eye docs have told me at various times.
I know you find the thick drops to be oily, but they can make a big difference esp if you are using a lot of the artificial tears.
What is the skin on the rest of your body like? Do you have dry skin? Any skin problems? I have psoriasis and because of that the eye docs check my eyes very carefully because it can increase the risk of complications with-any eye problems. Or so I am told. I have yet to see any studies about that, but both my skin and eye docs say this is so.
This next sounds weird. Bear with me. What color is your urine? Is it dark? You NEED to drink as much water as possible. The goal should be urine that is pale yellow or that you cannot really notice the color of. For every caffeinated beverage you drink, you also need at least an equivalent amt of water IN ADDITION to the normal amt you drink daily. Yes, you will pee like a racehorse. But it will help keep the general fluid level in your body up which helps keep your eyes hydrated. If you can't tolerate water, or can't handle large amounts of it, then drink non-caffeine drinks.
If you have problems drinking water, get a brita type filtration system. I do NOT recommend the ones that screw onto your sink. The biggest problem is having very little control over the amt of water that goes through them. If the filter isn't changed when it is full, it will dump whatever is in it back into the water as it comes out. It is very hard to monitor how many gallons come through those filters. With a pitcher style filter (and all brands are the same, brita, pur, any of them will do the job - even the no name one that is sold at walmart works as well as the brita), the mechanism to measure when it needs to be changed isn't super tecnical but IS more reliable than the faucet mounted ones. Esp the new ones that don't just have you put a date on the pitcher for when to change it.
I DO have a source for this info - husband. He spent more than 5 yrs doing water quality testing for a lab that reported to the EPA. He still knows a lot of poeple doing research on this stuff and those that can't afford whole house filtration systems or think it is overboard (because reality is that MOST cities do an excellent job of filtering the water - you get more problems from existing lead pipes, etc... in older homes), use the brita type filters for their families. And around here you can get a used pitcher at almost any thrift store at almost any time. then you just wash it and put a new filter in if you don't want to buy a new pitcher at the store. The pitcher is just the filter holder, mostly.
One awesome tip to get enough water without drinking a big glass at a time? Keep dixie cups (the small size) by every sink in the house. Every time you wash your hands, drink 1-2 dixie cups of water. Most of us wash our hands a LOT more than we think we do, and this is a great way to get and stay hydrated. One other thing that can help if you don't like or can't drink plain water is to add a bit of lemon, lime, orange or grapefruit juice to it. Just a splash. It adds a bit of flavor and changes the ph so that it is easier on your stomach.
Given eye problems, are you taking a vitamin with lutein? My mom's eye doctor (the md kind because she has had retinal problems, and several other severe eye issues) says that while it won't help short term, it does help long term. Also, get some of those granny sunglasses that go over reg glasses and have the panel on the side to block sun from the sides. We get a LOT of UV rays from the sides of sunglasses and these also can make a difference. They may not address the immediate problem but do offer some wind protection that reg glasses won't. Walmart sells them for $20 or less (solar shield is the brand I can find) and they are something that can ease the strain and every little bit helps.
Sorry this rambles. We have a range of eye problems in the family and these are what the docs have recommended to us.