You need to regrade the property to get rid of the ditches. That is best done with a tractor and a tool called a box blade. Landscapers all use them. It would take no more than a morning. A landscaper will bring in more soil if needed or remove soil if you have too much. When he leaves your soil will be ready for action. The drainage will be perfect and surface will be profiled to provide that drainage. You can lay the St Augustine sod down without doing anything else. And you might not have to kill the weeds before starting that project. Do you know what weeds you have? The really bad ones in the south would be bind weed, nutgrass, and kyllinga. Bermuda can be a headache but in your shade it will never be a problem.

If the landscaper comes on a Friday, you could easily have a lawn by Sunday.

Where do you live? Zone 9 is not helpful because there are zone 9s in many different states with many different soil and head situations.

With St Augustine it needs an inch of water per week in the heat of summer. What does not fall as rain needs to be supplemented with sprinklers.

Mow St Aug at the mower's highest setting for best appearance and drought tolerance. If it gets really hot and dry, then stop mowing until the heat breaks. Let the grass grow up tall and it will survive with less water.