Desvelo comes from the verb desvelarse, meaning to stay awake or to lose sleep. It's a more evocative word than simply saying "I can't sleep."
In the song, Tito Rojas sings, "Y por las noches si no estás conmigo yo me desvelo" (And at night if you're not with me, I stay awake). It beautifully conveys the depth of his longing and how his beloved's absence affects his peace, making it a memorable and emotionally resonant word.
"Siempre Seré" is a salsa romántica about loving someone you can only have in secret. The singer meets her again after a long gap, counting the hours they manage to steal, and he has no illusions about his place: he is the one who calms her longing and the tenderness that wakes her passion, but never the great love that first won her heart.
The chorus is where the ache sits. He would rather take the smaller part than lose her altogether, and he says it plainly. He will always be the comfort. He will never be the dream.
The long soneo section at the end is Tito Rojas answering his own chorus, line by line. He is the rain that wets her hair, the man who does not forget her, the one who forgives her when she has no love to give. It closes on a quiet call and response with the coro, a few words at a time, all the way to "hasta la muerte".