Trying to install cheap pigtails from eBay

Thank you for including enough detail. Yes, soldering will likely solve this. A better solution to the parts you found on eBay would be to visit a pick-your-own-parts salvage yard and get a good used fan connector with plenty of wire. That way you're certain the wires are of the correct size, and the plug and terminals will be correct. If the original terminals were overheated and the connector body was melted, both halves must be replaced. I wouldn't buy a used fan assembly for that. Instead, just cut the wires and plug from the fan and splice it onto your fan.

The first thing to watch for is as I just mentioned, if the terminals were overheated and melted, both parts must be replaced. Same if one of those parts is a switch. When overheated, the terminals develop resistance to electrical current flow. That resistance causes heat to be generated when current flows through it, and that heat leads to more resistance and more heat until the terminals burn up again. I've never had luck shining up darkened terminals. They still cause trouble down the road.

The next issue has to do with the different kinds of connectors used for splicing wires together. If you're familiar with "Scotch-Lok" connectors, those have no place in the automotive world. They're fast, but they don't seal out moisture. A lot of people use them when adding on a trailer wiring harness. No soldering or wire striping is involved, but they're easily the number one reason for inoperative trailer lights a few years later. By piercing the vehicle's original harness, moisture has another place to get in and cause corrosion. Any time I work on a vehicle for dead or intermittent rear lights, an added-on trailer harness is the first thing I look for. If I see that, I start my diagnosis with a visual inspection of the wiring under the vehicle.

Wire nuts are also not acceptable. It's a good idea to carry a few, but they're only good for temporary emergency repairs to get you home. The only proper repair under the hood or outside the cabin is to solder the wires, then seal them with moisture-proof heat-shrink tubing. I prefer to slide the strands from each wire into each other, then solder them. Twisting them together, like we do with house wiring, leads to a big mess that's hard to seal with larger gauge wires. Once I have the joint soldered, I use a small needle nose pliers to press down any sharp points that are sticking up so they don't poke a hole in the heat-shrink tubing.

Electrical tape is not a good choice for automotive use because it will unravel into a gooey mess on a hot day. If you look at what was used at the factory to neaten the appearance of wire harnesses, you'll see it looks like electrical tape, but it's really just "wire wrap" that is not sticky. You can buy rolls of that stuff at any auto parts store if you're concerned with appearance of the repaired area, otherwise nylon tie straps are just as effective.

If all you have is standard heat-shrink tubing, that is better than anything else for use under the hood. The preferred is the moisture-proof stuff with hot-melt glue inside, but it's hard to find that in a variety of sizes at hardware stores. Usually all they have is too large for automotive work. You should have better luck at an auto parts store. Chrysler dealers' part departments sell it in pieces long enough to cut in half to make two repairs.

There is something new on the market that can make life easier when you have to make a splice in a confined area where it's hard to reach. They look like a butt connector inside a clear plastic tube about an inch and a half long, and they're available for different gauge wires. All you have to do is slide the two wire ends into the connector, then warm it with a hot air gun. No need to reach in and crimp the connector first. Solder is already in there that flows into the wires when heated, then the clear tube is heat-shrink tubing. The only thing that can cause a poor joint, as with all other methods, is if the wires are not clean and shiny. Corroded copper wire has a dull brown color. That is a powder that prevents solder from flowing freely. Very often you'll find, even with a big messy glob of solder, the wires can be pulled apart. Scraping or sanding usually doesn't help. The best solution is to keep cutting the wire back until you reach shiny copper, and splice to that. Mitsubishi products in particular are well-known to have wiring trouble due to moisture, especially if the vehicle was in a flood. Seems the water gets in, then gets wicked through the strands inside the insulation. For this kind of damage, the only effective repair is to replace the complete wiring harness.

If you are not real experienced with soldering, I can help with that. One point that helps most people is understanding the solder flows toward the heat source. Also, the solder only is responsible for making a good electrical connection, not a good mechanical one. That's where sliding the strands of wire together forms the mechanical joint first, then soldering makes the electrical joint. It can help to touch a little solder to the tip of the soldering iron to promote faster heat transfer into the wire strands, then add solder to an area away from the iron. The wire should become hot enough to melt the solder a good quarter inch from the iron's tip, then the molten solder will flow through the strands and over to the tip. You can get away with adding the solder close to the tip, and it will flow some, but it won't cover as great an area, and will take a little longer to complete each splice. There's flux in the solder too to help it flow. When soldering "backward", meaning the solder is added next to the tip, the flux burns off before it has a chance to promote good adhesion as the solder flows away.

When you're done replacing wires, terminals, or connectors, be sure to check the fan spins the right way. If it spins backward and blows air forward, it can be easier to remove the two terminals from the plug, and switch them, rather than going through the work of doing the job over. Also check that as the car goes up and down over bumpy roads, the wires won't hit the fan or hot engine or exhaust parts. The wires should be anchored to prevent excessive swinging. Wires that are free to swing can work-harden and crack over time, and they're susceptible to ice forming on them in winter that weighs the wires down, possibly leading to them pulling apart.

Please keep me updated on your progress, and let me know if you need more wondrous advice.

If a connector terminal is getting too hot, it is not making a good, solid connection or it's too small for the amount of current it has to pass. If a wire is getting hot, especially a new one, it also is too small.

This is kind of a hard circuit to work on as far as reaching test points, but if you can get the fan to go dead and stay that way, that's the perfect time to take some voltage readings. With those we can figure out exactly where the problem is. This is the diagram for the fan and relay. If you can get to the connector terminals to take the readings, you should find 12 volts at the red arrow when the fan should be running, and 0 volts at the blue arrow all the time. Any voltage at the blue arrow indicates too much resistance in that wire or terminals.

As for the direction, the fan should blow air toward the engine, same as the natural air flow when you're driving. If it's blowing forward, the two wires have to be switched in one half of the connector. Every connector is purposely made confusing to take apart. Look for a little "pimento", or lock that can be pried out in the middle, between the terminals. Once removed, that will give you access to a little plastic finger that must be lifted in some way to let the terminal slide out. These can be frustrating to figure out. The connector views are at the bottom to show which wires are involved. Let me know if you can't read them. I can cut them apart, then expand them in sections.