So, I recently moved. In my old place, I used my Nest without a common wire and never had an issue. However, in my new place, my Nest thermostat was working fine for about 20 days and started losing power. I put the old fashioned thermostat back for now. I know I need a C wire. My thermostat has 3 wires for the AC. I was thinking of buying a 18/2 thermostat wire and just fish it to the thermostat. That's the easy part. I take pride in doing DIY but I'm not so sure when it comes to electrical components. My next question is once I fish the new thermostat wire, what do I do next? I will attach pictures for reference.
It seems that non-standard colors and mismatching connections are everywhere, so confirm everything and don't rely on color just because that's what the wiring diagrams say. From all of your diagrams, it appears there are several locations that a common wire should be accessible.
On the simplified diagram posted first, the second wire running to your A/C is common (brown if the diagram can be trusted). On the more detailed wiring diagrams, it indicates that yellow/black are connected to the primary side, while red/brown are connected to the secondary (24VAC) side of your transformer (I can see those colors in your pictures). In this case brown, again, is your common. If we follow this back, there should be a C terminal on a PCB where the G and R wires connect. If you can find it, I would connect your new C wire to this PCB location. You'll only use 1 wire from your new 18/2 bundle.
Formerly, your Nest worked by leaking power through the furnace or air conditioner's operating relay. The relay was old and brutish enough that it flowed enough current for it to work. Now, you have an electronic control furnace that does not have those smarts. This is similar to the problem people have with lighted switches or dimmers when they switch to CFL/LED lights.
And there is a similar solution available. Put a device in parallel with the furnace or A/C control which leaks enough current to act like the big old relay. This can be done with a power resistor, or a capacitor since it is AC. We don't do product recommendations, but the HVAC market should have products (UL listed) made for this. I would not try to wing-ding it yourself with electronic components (ЯU listed) unless you know exactly what you are doing.
"parallel" means the device would go between R and C down in the furnace.