No turn signals?????
Next, pull the radio to get at the DRL module mounted behind it. Test for flashing power at the dark blue with white (right) and light blue with white (left) wires when the ignition is on and the turn signal switch is activated. This will tell you if the switch is working.
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It is quite common for the front sockets to corrode beyond repair. They are mounted in the bottom of the light housing - right where water will collect. That's probably why you have a replacement socket on one side. However, it's not possible for the socket (or bulb) to allow the DRLs to work without the turn signals also working - they use the same filament in the dual filament bulbs. For example, you say the right front bulb is burned out yet both DRLs work... that's electrically impossible.
The left side is puzzling. I can't think of any scenario that would cause all the bulbs to go on but not flash on one side. Are both bulbs in the left rear working? Any burned out bulb (not just the front) will cause the non-flashing problem on that side.
Did you test the wires at the DRL module as suggested earlier?






