Newbie - Busted Taillight and Taillight connector

Hello,
I had a cracked taillight few months back, and few weeks back, my car started behaving werid with tailight warning. I changed the light bulb and after few days, it stopped working again.

When I checked, the connector had melted.

So I went ahead and ordered these parts:

QTY Part Number Description
1 63 21 7 165 865 Bulb Carrier for Taillight Left NOTE: This carrier holds 4 bulbs.
1 63 21 6 946 533 Taillight with Yellow Turn Signal for Fender Left
1 12 52 7 519 956 Connector Socket for Bulb Carrier Fender Taillight Left/Right (2 Per Car)


Now that I wanted to change the taillight I searched on how to do it, and found out that this apparently is a known issue.

How does one go about changing the connector? removing the wires from the connector and putting it into the new connector?

I thought the connector would open up (like a book) where I would align the wires and snap it back in place, but that doesn't seem to be the case.

Thought I would start a new thread, as I'm replacing the lens as well as the indicator issue, so could get help on both.

Any help, suggestions, guidance, tips?

Disclaimer:
This is my first DIY