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Yes, washing it would be a good idea. Make up a spray bottle with a solution of 1 teaspoon mild dish detergent to 2 cups water. Spray all over the plant, especially up onto the underside of the leaves, and down into the nooks and crannies in the center. Let it sit for 20 mins or so, then wash it all gently with a hose. Looking through a wide variety of rhipasalis info, I can't find any disease that produces "weeping sap." I think it's almost sure your plant has some kind of sucking insect - some scales or aphids are almost invisible, and it doesn't take but a few to start dripping that honeydew, which is easily mistaken for sap. Wash the plant in the same way, at weekly intervals for 4 weeks, and you should eradicate the little buggers.

What you have done is fine. This should help the growth, and recovery. Here is an article for more information: https://www.gardeningknowhow.com/ornamental/cacti-succulents/rhipsalis/rhipsalis-mistletoe-cactus.htm

You can prune these with very little worry. They respond well to pruning. You can, even, use this cut portion to start a new plant. This article will help: https://www.gardeningknowhow.com/ornamental/cacti-succulents/rhipsalis/rhipsalis-mistletoe-cactus.htm

Yours may be getting too much sun, that can darken foliage. Try moving it to a bright location without direct sun. If the leaves are dropping, you may have a root rot problem. In that case, repot as soon as possible in fresh succulent soil and clean container.

Here is an article about care for the general rhipsalis group.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/rhipsalis/rhipsalis-mistletoe-cactus.htm