Most of the dramas I'm about to recommend are all oldies from the late 90s / early 2000s, but total goodies that are worth the watch if you're looking for decent plots and heart tugging romance about bad boys turning good with the help of a loving FL.
Lovers - ML is a gangster & technically in a relationship when the FL comes along. She falls for him anyway and supports him through thick and thin, even though he feels he's not good enough for her because of his lifestyle. One of my all time favourite dramas!
Pride - Both the FL and ML suffer from abandonment issues, but the ML seems particularly affected. He carries around a lot of complicated feelings, shame being one of them. FL is pretty scarred too, but on the surface she appears to be coping better than him. They heal and love each other. This drama is amazing!
Cheap Love - ML is a denigrate who recruits women to act in porno flicks. Now if that isn't the most lowlife profession (being only slightly better than a pimp) one can have, then I don't know what is. ML knows he's scum and feels shame. FL loves him anyway, for reasons only she alone - at first - understands. I ended up loving the drama more than I thought I would. However, it unfortunately ends tragically.
I didn't see Always on your list. This is a movie, but matches what you're looking for perfectly. ML is a former boxer with a troubled past he's not very proud of. FL loves him unconditionally anyway. This is a classic and a must watch for all romance lovers.