Giving your plant little tiny sips of water is doing more harm than it is any good. Tap water has dissolved minerals within it. When you give your plant a small amount of water, none of it drains through the soil and out the drainage holes. Your plant soaks up this small amount of water and the dissolved solids get left behind in the soil where they eventually begin to accumulate, more and more, each and every time you give your plant another little sip.

When you water any plant, always flush out the whole entire pot with more than enough water. This washes away any excess mineral buildup from the soil. "Overwatering" a plant isn't about giving your plant too much water. That's where most novices get confused. It's about having a soil that holds on to too much water and remains soggy. Even though plants take in carbon dioxide thru their leaves, they also breathe in oxygen through their roots. And having a soggy soil with too much moisture essentially cuts off this supply of fresh air to the roots, which ultimately leads to those roots suffocating and rotting away.

I stopped typing to give your photos a much closer inspection and I noticed the overturned wine bottle sticking up out of its pot. Are you using this as a water globe? If so, then it's entirely unneeded. Ponytail palms store water in their caudices (caudex/trunk). Compound this wine bottle/water globe with what I suspect is a glazed pot with no drainage, and I think we have the answer to at least one of your problems. Glazed pots are completely fine, just so long as they have adequate drainage. It's what I have all of my ponytail palms planted in. Well, all except for one (long story). I would advise you to repot it as soon as possible into a pot with drainage holes, as well as with a fast draining soil mix. For mine, I use the Miracle Gro brand soil for cacti and succulents and it works fine. I don't even add anything to it, either. And when you water it, always flush out the whole pot. If other members on here saw how much I water mine, they would think I was crazy, and that I was drowning my plants. But they're all kept outdoors down here in the South, where it gets very hot and very humid. Plus, they all are planted in a soil that never gets soggy, even when I "drown" my plants. Because all of the excess water always drains away.

Now, about the leaves...

Your ponytail isn't getting enough sunlight. This is why the leaves hang straight down. These plants are notoriously tough and remarkably tolerant of less than ideal growing conditions. But even they can endure only so much. Just because a plant can tolerate low light levels, that doesn't mean that it prefers lower light levels. I would suggest gradually moving it to a much brighter window, or brightly shaded porch. All of mine grow in full, direct sunlight almost all year long, with no shade whatsoever. But they are used to receiving that much sunlight. Yours isn't. It would need time to acclimate and adjust without getting sunburned.