The NTSB report is a great resource when looking for information about an incident like this.
There is an Engine Dual Failure Checklist discussed starting in section 188.8.131.52 of the report. This includes steps to attempt restarting the engines, and further steps depending on whether or not the engines can be started, and finally steps to help prepare for a forced landing. Pilots train on engine failures regularly, so they are already familiar with the procedures. The report discusses how their actions compared to what the checklists say. Since the incident occurred at such a low altitude right after takeoff, they didn't have time to complete all of the steps. They did what they had time for, and had to make quick decisions about what their options were.
The report also discusses in section 184.108.40.206 the ditching training the pilots received. There was guidance in the manual, but no specific scenarios included in their simulator training.
The autopilot is great for use in normal situations. However, as soon as anything is going wrong, the pilot should take control of the plane. You do not want to take time to figure out "what's it doing now" in a difficult situation. So in this situation, the autopilot probably disconnected when the engines went out, or was manually disconnected by the pilots.
Section 1.6.3 talks about the flight envelope protections. I believe these protections are what you are referring to when you say that the autopilot has more control. They are designed to protect the plane from stalling or otherwise exiting controlled flight. Because the pilot was able to start the APU to provide electrical power, the plane remained in the "normal law" mode where these protections were available. This limits the control the pilot has in order to protect the plane, in the sense that the plane will override pilot inputs to avoid dangerous situations.
Section 2.7.2 discusses the impact of these protections in this case.
The airplane’s airspeed in the last 150 feet of the descent was low
enough to activate the alpha-protection mode of the airplane’s
fly-by-wire envelope protection features...
Because of these features, the airplane could not reach the maximum
AOA attainable in pitch normal law for the airplane weight and
configuration; however, the airplane did provide maximum performance
for the weight and configuration at that time...
The flight envelope protections allowed the captain to pull full aft
on the sidestick without the risk of stalling the airplane.
An interesting point in the report is that Airbus certified the plane to be able to land on the water intact under certain conditions. Here is how the actual conditions compared to the certified conditions.
The report mentions that it would be extremely difficult to meet all of these requirements in an airplane with no power. The values that are higher, notably the mass and descent rate, are factors in the damage the airplane received.