Unfortunately, there are not many accessible overviews of Mesopotamian divination and magic. The best introduction is Stefan Maul's The Art of Divination in the Ancient Near East, which is a bit pricy ($60 on Amazon). If that's too expensive, see if your local library (once it reopens) can get you a copy via interlibrary loan.

Also take a look at Religions of the Ancient World: A Guide edited by Sarah Iles Johnston, which contains a chapter on divination and prophecy.

In the meantime, a few free online resources:

• Mesopotamian Astrology: An Introduction to Babylonian and Assyrian Celestial Divination by Ulla Koch

• "Divination culture and the handling of the future" by Stefan Maul, excerpted from The Babylonian World edited by Gwendolyn Leick

• "The Ancient Near East" by Daniel Schwemer in The Cambridge History of Magic and Witchcraft in the West edited by David J. Collins

• "Magic Rituals: Conceptualization and Performance" by Daniel Schwemer, excerpted from The Oxford Handbook of Cuneiform Culture edited by Karen Radner and Eleanor Robson

• "Sheep and Sky: Systems of Divinatory Interpretation" by Ulla Koch, excerpted from The Oxford Handbook of Cuneiform Culture edited by Karen Radner and Eleanor Robson

• "The arts of the diviner" in chapter IV of Oppenheim's Ancient Mesopotamia: Portrait of a Dead Civilization

Also see the relevant translations of omina and oracular inquiries in the State Archives of Assyria Online.