Water bottles are a total pain! But if you can find one that doesn't drip too much and lets enough water out I believe they are the safest option. With a bowl the water can get contaminated too easily (so it shouldn't be on the substrate but on a shelf) and hamsters can also fall in them and get wet and get sick. So it's not a total no-no if you know what you're doing, but a bottle is much much safer - two bottles ideally in case one stops working or empties itself. I've seen people in the US say these are the best ones for working well and not leaking and they seem very popular with lots of excellent reviews
Amazon.com : Kordon/Oasis (Novalek) SOA80304 Bell Bottle and Hold Guard Small Animal Value Set, 4-Ounce, variety colors : Pet Water Bottles : Pet Supplies
I tried a glass bottle once - I like the idea, but they are so heavy and ours kept falling over
Which really upset our hamster each time and was a bit dangerous if it happened while he was drinking from it as the spout would twist round. Obviously I hadn't found a way to fix it securely enough!
If the cage is getting wet at a different part of the cage to the bottle, then it's probably wee. As the others say, she could have diabetes, so no sugar at all (a hamster mix shouldn't have any dried fruit in as that is also high in sugar). The best treats would be fresh food like a bit of cheese or nuts or sugar-free hamster treat biscuits or something, or seeds, like a pumpkin seed or sunflower seed.
If her eyes are closed it does sound like she is really not well or they are gummed up, even if you can't see any crustiness.
Be gentle with her - it must be so stressful for her if she's not well and can't see.
How much depth of substrate do you have? It will make it easier for you and her if you have it as deep as you possibly can, so you can spot clean the wet areas without removing all of her scent, and so you don't need to clean the cage base each time.
I seem to remember you haven't had her long - you were posting about taming I think. Where did you get her from? I think Pets at Home in the Uk have a vet service which people can use if the hamster has recently been bought (ie they may have already been sick when bought). If not, then she would need to see a vet. But as you say, observe her over the next 2 or 3 days. Make sure she has lots of torn up white toilet paper for nesting material (not kitchen roll), and maybe try not to handle her for a few days to give her chance to relax a bit (or poke her with a chopstick either!). You can talk to her gently through the bars though as your voice will get more and more familiar and be reassuring for her. Also you could do the tissue trick with the toilet paper - put the sheets of toilet paper up your sleeve for a few hours, then tear them into strips and put them in the cage in a bit pile somewhere so she can pouch them and take them to her nest, and she will get familiar with your scent.