Engine running very hot but plenty of coolant - is it related to recent a/c work?

when u turn your ac on do the radiator cooling fans come on or if they are already on do they speed up after a few seconds when condensor heats up the cooling fans should run at a low speed to cool condensor when engine gets to running temp opproxamatly222 degrese fan should cycle between low speed and high speed.this is a very proper answer would like if u would email the problem to me once fixed thank u

tryin to find an answer to a very similar problem. replaced compressor,dryer,cooling fan and runs fine now but when air is off and fan cuts on the temp gauge jumps from normal to 1st hot line...then cools back down...wondering if air in system.