Per the Farmer's Almanac Jade Plant Guide, it seems like your plant isn't getting frequent or deep enough watering:
If the plant starts to drop its leaves, if leaves start to shrivel, or
if brown spots appear on the leaves, it is an indication that the
plant needs more water.
So give it a nice, deep watering, but the water shouldn't spill out of the tray below it. Then I would give it another watering not long after that- a few days later. The soil should be moist, but not sopping, drowning wet.
After those two waterings, maybe a third watering to catch the plant back up to a nice state, you should follow the watering instructions in the guide above.
Your black thumb may not be as bad as you think, especially if you're starting with stressed plants. Your inclination to do get rid of the shriveled and drooping branch is spot on. Because it appears that the drooping and shriveled leaves come from one woody stem, I would simply remove the whole stem from the pot. Loosen the soil around it, try to wiggle it away from the other stems and remove it. Put the soil back. Cutting it would also get rid of the unsightly branches, but you would be left with a useless and dead stem in your pot.
You are doing the absolute right thing by putting the plant in the south window- another indication your thumb is turning more green! It looks leggy and rather tall, not full and robust near the base as is desired and that comes from not having enough sunlight. It's likely the previous person added the blue stake in an attempt to help the leggy and drying growth from tipping over. A healthy jade plant does not require staking.
By placing it in the southern window you should be addressing the leggy problem for future growth. I'm not all that bothered by leggy plants, but if you wanted to create the typically desired look, you could cut the tall branch to encourage new, lower shoots. You could also take some leaf cuttings and encourage new plants and put those in the pot and then you would have additional new, lower growth. I encourage you to experiment with the leaf cuttings- the guide gives instructions for this process. It is incredible to watch a new plant come to life from a little leaf! It can be fun, your plant will be fuller, you will be empowered, and it can give you increased confidence ;) What's not to be gained?