Penando comes from the verb penar, which means to suffer, grieve, or endure hardship. While sufrir is a more common verb for 'to suffer', penar often carries a deeper, more prolonged sense of sorrow or struggle.
In the song, the lyric "Tanto que anduve penando" (So much that I walked around suffering/grieving) beautifully conveys the singer's long period of emotional pain and searching before finding solace. It's a more evocative and poetic choice than simply using sufrir.
"Abrázame" starts with a search, not a scene. Ángela Aguilar sings "Me he preguntado" [I have asked myself] and "Dónde le puedo encontrar" [Where I can find him], waiting on a sign before anything else happens. The relief, when it comes, is stated flatly: "Mi sufrimiento ha acabado" [My suffering has ended], because "Desde que veo tu mirar" [Since I see your gaze] is the whole explanation the song needs. Felipe Botello y El Sonoro Rugir's horns and strings never let the arrangement go quiet, so even this line lands at full volume instead of as a private confession.
The title line is the ask underneath everything else: "Abrázame, abrázame fuerte" [Hug me, hug me tight], followed straight by "No me dejes, por favor" [Don't leave me, please]. By the closing minute that second line repeats four times running, and it stops sounding like a request and starts sounding like someone who isn't letting go. The detail worth catching on a second listen is "Corazón pardo" [Brown heart], next to "Por arriesgar y fallar" [For risking and failing], the fear she took a chance on, sitting right beside the relief of it working out.