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I don't believe it will completely kill them, although it might hinder the flowering process. At most you will just have to wait another year for the blooms to come back.

Make sure your plant is getting enough light. Low light could cause they yellowing leaves and droopiness. Also check that it is planted in very well drained soil. If the soil drainage is poor, the bottom of the pot could be staying wet even when the top of the soil is dry.
https://www.gardeningknowhow.com/ornamental/bulbs/amaryllis-hippeastrum/drooping-amaryllis-leaves.htm

Here is a plan for helping an amaryllis rebloom:
http://www.usna.usda.gov/Gardens/faqs/AmaryllisBloom.html

They way mine have always done, has been that they will flower within a few weeks of each other. As long as they do not get chilled, they should last. Not all of them will but there should still be some blooms left on it by that time. There is not real guarantee that this will be the case, but with proper care it is a real possibility. This article will help you with the care of these: https://www.gardeningknowhow.com/ornamental/bulbs/amaryllis-hippeastrum/amaryllis-care-instructions-how-to-care-for-an-amaryllis.htm

Your original idea is best. Removing and drying them will help keep them better, but they can be wintered in container as well. Don't resume watering. Let them continue into dormancy. Here is an article that will help: https://www.gardeningknowhow.com/ornamental/bulbs/amaryllis-hippeastrum/amaryllis-bulb-storage.htm

The only time that I am aware of this happening is with overly wet soil. It will exude a sap to try and correct the moisture around its root zone. The plant doesn't tolerate wet soil very well. Only moist. It would help to have a picture to see the conditions. Make sure that the pot it is in has proper drainage. If it has a catch pan, then there is likely standing water that is not visible to you, even if the top doesn't appear wet. Just make sure there is proper drainage in the pot, and keep an eye out for root rot. This will show as brown or black leaf tips that will consume the whole plant eventually.

This article will help refresh on the care of these plants: https://www.gardeningknowhow.com/ornamental/bulbs/amaryllis-hippeastrum/amaryllis-care-instructions-how-to-care-for-an-amaryllis.htm

You don't need a special glass vase for amaryllis. Any tall vase that is large enough to accommodate your bulb is fine. By being tall, it helps the plant stay upright. River or decorative rock or stones go in the bottom where water will be. This also adds weight, again to help keep the plant upright. Bulbs bloomed in water probably won't be able to regenerate the bulb for a re-bloom next year. If you want to maintain the bulb, try transferring it into a potting mix until the leaves die down. I don't recommend the smallish glass containers that hold the bulb above the water. They are too likely to tip over. You can see what one company does w/ glass vases for amaryllis at. https://www.whiteflowerfarm.com/amaryllis-glass.html