Is the marriage of a woman, which took place after a previous relationship with her husband and before their repentance from this relationship, valid Islamically, and are they considered persistent adulterers?

The marriage of an adulteress is a matter of scholarly disagreement. The opinion of Imam Ahmad, which states that it is impermissible except after repentance, is the one given fatwa (religious legal opinion) upon. However, in this particular case, since you are not certain of the occurrence of zina (unlawful sexual intercourse) in its sharia-defined meaning, your marriage is valid by consensus, as Islamic rulings are not established by doubt. As for metaphorical zina, such as touching private parts, it is not the actual zina that necessitates a hadd (prescribed punishment). The questioners have repented to Allah, and Allah accepts repentance from His servants. Even if complete zina had occurred before the marriage contract and procreation, the school of thought of the three Imams lifts the burden in such a case.