Without an annulment, is it OK to date as long as we are chaste?

No. Currently you are not even legally divorced, much less have you received a decree of nullity from the Church. Until the latter happens, you must presume that you are a married woman and may not date anyone. Once you are legally divorced, you will no longer be married in the eyes of the state, but you will be married in the eyes of the Church unless and until you receive an annulment—and there is never any guarantee that an annulment will be granted. If you maintain a life of chastity appropriate to your state as a married woman legally separated from her husband (the Church considers civil divorce the equivalent of a legal separation), and otherwise remain in a state of grace, you may receive the sacraments. Once an annulment is granted, then you will be free to date.