I feel far from worship when I have my period. What practices are still open to me?

Menstruation does not cut your connection with Allah; many doors of worship remain wide open. Increase dhikr, seeking forgiveness (istighfar), sending blessings (selawat) on the Prophet and supplication at all times. Listening to Qur'an recitation and religious talks, giving charity, and doing good and serving your family are all rewarded. Fill the times usually spent in prayer with dhikr so that your spiritual routine is not broken. On the ruling on touching or reading the mushaf during menstruation, there is indeed a difference of opinion among scholars; for the right practice in your situation, consult a trusted teacher or the religious office. Make this period a time of rest that is still spiritually productive. See fiqh of menstruation for couples.