Headlights not working properly

How about some duct tape and a flashlight! No you want the real fix huh.

Okay at the plug for the light you should have the two orange wires, these are battery power feeds through the ext lamps 10A fuse to both high and low beam.
Those go to pins F and H, both are the same so does not matter which goes to which.

The dark blue wires are a pain as you need to test them to find out which one of them is the low beam ground and the other will be battery voltage to the turn signal. The easy way to do that is with a test light. Connect one side to ground, turn on the right turn signal with the headlights off. Touch the two wires one at a time, the one that makes the light flash connects to pin D and the other dark blue connects to pin J.
If you do not own a test light, you have one right on the car. Just be sure that the ground (pin A) is connected, it grounds both the turn signal and the marker light. Now just touch the blue wires one at a time to the wire on pin D just like you would use the test light. The one that blinks the light is the turn. The one that does nothing goes to pin J.

Then there will be a pink wire connected to pin G. That is the high beam switched ground, but it is switched through the DRL relay. When the DRL's are on, both high beams are grounded in series which halves their light output. When the DRL relay is turned on it switches the grounds to both lamps on independently and you get regular high beams.

Oh and if you look at pin E it says that is power from the dash. That is the marker/parking lamp feed. It pulls power from the dash lights so that if you are driving along and the dash suddenly goes out, you know you also lost the park/tail/marker lights.

Hope this provides a lighted pathway.

Either the wiring out to the plug is damaged or there is a blown fuse in the system. How did the connector get damaged? Accident or? If it was pinched between something it is possible that it is damaged in more than one spot.

The low beams are powered through the orange wires. The switch grounds them using the dark blue wires to turn them on. If each one has power on the orange wire. Then grounding the dark blue that comes off the other terminal should turn them on. That wire should also show a ground with the low beams on.

It is possible to get things mixed up because they also used dark blue for the DRL feed to the high beams on one side and a signal on the other.

The orange wires are power feeds. On the right hand (passenger) side there should be two orange. One feeds high beam, the other low beam. Then a pink which is the high beam ground, and the dark blue that is the low beam ground. The dark blue is connected across the car to the low beam on the drivers side. If you need to figure out which is which you could pull the other headlight out and look for continuity between the lights.

The orange wires are power feeds one for high one for low. If the low is not getting power check that those are connected and that the fuse (DRL/EXT lamps 10A) is still good. It powers the passengers side and DRL system.
It sounds like you just need to sort out the blue wires. These diagrams with the colors on it and showing the entire system might help. The first is the headlights only and the second shows the turn.

You say the high beams both work. So the headlight relay is working as it is what sends the ground point to the dimmer switch.
Then it comes out of the dimmer on the brown lead which connects under the dash to the harness with the dark blue wire and that goes out to the headlights. It provides a ground only with the dimmer switch on low beam. It shows that the two blue wires are spliced together in the harness near the external temperature sensor (which is on the rad. Core support).
The easiest way to find it would be to use a test light connected to battery positive. Now for a different test method to keep the power/ground issue apart. Remove the headlight relay. Use the test light to find the contact that has power. That will be BC on the diagram. Then the one diagonal to that is the other side of the coil (AA on the diagram). The other two pins are BA which will be direct to ground and pin AC which is the ground to the dimmer switch. Insert a jumper wire across those two pins. Now put the dimmer switch on low beam, Leave the key "off" and the headlight switch "off". Now if you use the test light connected to battery Positive there will be one blue wire that is grounded. Unplug or remove the low beam bulbs. Take the test light and you should find a ground connection on the dark blue wire on the drivers side. Then find the dark blue ground on the passengers side.
Now for the other part of the test. Because the wire should only ground through that relay and the jumper you installed. Remove the jumper, now connect the test light to battery positive. Then touch it to the same two terminal you just removed the jumper from. It should light up on one (BA) and not on the other. If it lights up on both you have a short in the blue wire. That would explain why the one side stays on and the other side does not work, if the wire is broken and grounded in the wrong spot.

They are normally tied together so you could simply run a new wire across to make it work.