Hey there, I corrected what you wrote at the very bottom.
All the words are used as frequently as the others, none are "formal", some are just more descriptive.
I would use "slump", personally, as something like ; "I'm in a slump" Meaning, I am feeling down, unmotivated, in a bad place mentally. It could mean you are not doing well in your job, it could mean you are depressed, it could be many things. You could also say it how you wrote it, but more often it's used as a noun (the way I did)
Sprawled you have correct. When someone sleeping next to you gets out of bed, and you have the whole bed to yourself suddenly, you usually sprawl out!
Slouch is bad posture and you have it correct. My mom would always tell me "Stop slouching, Nicki!" and I would straighten up.
Sink is correct as you've listed it. I hope you also realize it could mean to fall beneath water. If you drive a car into a lake, it will sink to the bottom. Here, it means sink "figuratively" because the person cannot ACTUALLY sink into the couch, it is just a way of expressing how he appears.
So, someone without energy is probably in a slump. They might be slouching, they might come home and sink into the couch. I have come home before and been so tired that I've simply sprawled out on the floor! All words can work.
Corrected :
He was weak so he slumped into the sofa
..so he sprawled OUT on the sofa
..so he slouched ON the sofa
..so he sunk into the sofa
Hoping I helped :)