This vehicle is overheating and has no heat?

I checked the coolant level and the oil level, and they were fine. When you drive the car, it drives fine for five minutes or a couple miles, then the temperature gage starts to climb until you get a message that says the Engine is hot and overheating. It also says no traction control. And it tells you to shut the car off immediately. When you turn the car off, you continue to hear the fan running for several minutes trying to cool the engine down. My first instinct was that the thermostat was bad, so I replaced it. I was worried about adding coolant and just starting the car, so I did a lot of research on how you refill the radiator and the coolant reservoir after draining the system. So, when I started the car, I left the radiator cap off and filled the radiator with coolant and then filled the reservoir up to the fuel line. I was also trying to get any air bubbles or air pockets out. After letting the car run for a few minutes I put the radiator cap back on. When I tried to drive the car, it did the same thing as before and after a few minutes or a mile or two it overheated and said to stop immediately. Everything I’ve read says in these situations it’s typically the thermostat, the fan, the relay, or the water pump. The thermostat has been replaced. The fan is running even after you shut the car off trying to cool everything down. The relay tells the fan when to come on or so I think, so I don’t think it has a bad relay. So that leaves me with the water pump. I just don’t want to replace the water pump unless that is the cause of the problem.