Are all US fighter/combat aircraft constantly armed, even just in training? I remember hearing once upon a time that all US fighter/combat aircraft constantly armed, even just in training. If I recall correctly, it's because the pilots need to be ready at a moment's notice.

Is this true?
As a general rule, unless you are going to use ordnance e.g. training, alert status, or war, you are not going to have it loaded on the jet.  It's just another risk to personnel or equipment out on the line, in hangars, etc.  Aircraft returning with live ordnance are disarmed and unloaded after shutdown and the weapons defused and returned to the bomb farms on base or magazines aboard ship.