Canon EOS 5D Mark III
Why you using Program Mode on a Top DSLR. Don't make sense to me. Sorry. Try "A" instead.
I assume you mean "try "Av"" mode. A mode (strictly, green-A+-with-a-box-round-it mode) is indeed an idiot mode put there to allow the camera to be used by those who have bought it as bling. But do not dismiss P mode which in undemanding situations will give a perfectly sensible balance of aperture and shutter speed. The trick is to know when not to use it rather than avoiding it completely.
The OP's problem presumably arises because the camera had already seen some use and had been customised to take AF off the shutter button before the OP acquired it. The answer is to use the menu options to clear all custom functions and restore the camera to its default settings.