Wondering why my 2005 GTO is overheating and isnt blowing heat after2thermostats

gmc4me: I'm having the same issue as goat05. I drive a 06 gto and yesterday, all of a sudden the temp gauge shot up to "H" and the system alert went off. I then noticed the heater wasn't blowing cold air, even when the car had been at operating temp for quite some time. I went ahead and changed the t-stat, drained all coolant, filled everything back up to appropriate levels and burped the system for about 30 mins. During the burping, the first 20 mins the car was at the right temp (right b/t 1/4 mark and 1/2 mark on the gauge, where it's always been) then all of a sudden the temp started to rise again and fall again. I went and checked the hose that runs from the radiator to the water pump and noticed i didn't feel any water running through and it was semi-cool to the touch. If coolant was constantly flowing through the hose, you wouldn't be able to hold on too long correct? I'm guessing the water pump has gone out, but looking for feedback before I start the process of replacing the water pump. *Also, during the burping phase when the radiator cap was off, I noticed the water wasn't constantly moving through the system. I would stay at the same level for a long period of time. Is it also possible the water pump could be clogged up? I've read other threads that the Dex-One coolant can build up and gunk up over time. Any feedback would be greatly appreciated.