Both are correct. There are some verbs in Greek called "συνηρημένα ρήματα" (contracted verbs) that have two ways of telling them. Verbs that end in -αω (such as αγαπάω) fall in this category. Εγώ αγαπάω/αγαπώ, εσυ αγαπάς, αυτός αγαπάει/αγαπά, εμείς αγαπάμε/αγαπούμε, εσείς αγαπάτε, αυτοί αγαπάν(ε)/αγαπούν(ε)
Would "The woman loves her husband" be an acceptable translation?
No, because there is no possessive and no indication of their relationship. her husband = τον άνδρα της.
In this listening exercise the voice says "άνδρα" while the word is rejected as a typo in my answer.
Why does it suggest to write "love" in the translation, and then it shows the answer as incorrect and says that the correct translation for "αγαπα" is "adore"?
When translating English to Greek, yes, but not in a Greek listening exercise ("type what you hear").