I get excited about words like this because they carry a whole life of usage in one short sound. For 'awwāb' (often transliterated as awab or awwab), I usually think in terms of motion and repetition rather than a one-off mood. A clear, literal rendering is 'frequently returning (to God)' or 'often returning to God' — that keeps the verb sense intact and is faithful to the Arabic intensive form that implies repeated action.
For more natural English I often prefer 'constantly repentant' or 'frequently repentant' in theological or devotional contexts, because those phrases communicate both the emotional state and the repeated practice of repentance. If I'm translating poetry, though, I like the slightly more lyrical 'ever-turning' or 'ever-returning', which preserves the movement and sounds nicer on the ear.
If the word appears as an epithet or name, 'the Oft-Returning' works well in older-style English and gives a noble, almost titular feel. Choosing among these depends on register: literal, devotional, poetic, or titular — pick the one that matches the tone of the passage in 'Qur'an' or hadith literature where it shows up.