"тоска", mysterious as it is, is associated with melancholical, kinda depressing yearning for thing to be better/more in line with what you want. SO it may mean two things: "yearning/depression" and just "boredom" (then it is the same as "скука")
If what you mean is a strong, emotionally "painful" desire for something that is, perhaps, out of your reach, and has been so, then "тоска" is kind of a right word. "Заглушить тоску" will work (though, I just made it up, and it sounds as a "temporary" solution).
Желание is "wish, desire" and "стремление" is more like "striving, aspiration" and even "intention" in some context, i.e. something that makes you strive for something (whil "тоска" is a passive thing)
So.. what is the meaning of your "quench the yearning"? Is it the same as "quench your thirst" (satisfy it) or "quench the disire" ("kill" it). Or maybe "shut off" your desire if only temporary?
The alternative words for profound yearnung:
- острое/сильное желание
- жажда (lit. "thirst", a strong and sometimes active desire)
- стремление (if it is active! And only then)