I would advise against trying to make this sound legato without pedal. This is hard even for an advanced pianist, and the risk is that you create a lot of tension trying to hold the notes down as long as possible.
My recommendation is this: first practice the left hand alone, with pedal. Wrist and fingers should be as relaxed as possible, with a featherlight touch. You should be lightly caressing the keys. The pedal doesn't need to go right to the bottom, just far enough to let the dampers just clear the strings, and it shouldn't come completely up to the top either: each pedal change is a tiny movement.
