The Army's pistol marksmanship and unit level leadership by and large are not what I would count on for advice or competency with regards to pistol marksmanship. Our pistol marksmanship program is abysmal. It is getting better but it will take a long time to change the culture but even then you still have to factor in the lowest common denominator. I would however use my safety, as much as possible. If you are worried about remembering to flip the safety off and firing from retention reread what Maple Leaf Actual stated. Pectoral index is retention so if you teach yourself to draw and remove the safety at pectoral index you will be able to fire from retention out to full extension. This requires no live fire to train and ingrain. You can do this just standing outside the arms room in a safe area.
If you decide to run and move with the gun off safe, I would highly advise against it. It presents a safety risk to you and everyone around you, even if you are good about keeping your trigger finger in register. We are all human, the pistol is more or less an extension of your arm and is much easier to point at stuff you don't want to than a carbine or rifle. Therefore I am and always have been firm in the camp of sights on safety off, sights off safety on for everything but admittingly, there might need to be some flexibility where hand size and ergonomics play a large factor in reaching the controls on a pistol. The P320/XL/365 safety leaver is absolutely F'ing useless in terms or size/location on the frame but you can still safely work around this while drawing, moving and reloading.
In spending a lot of time on this platform as of late I believe that the best way to run the 320, 365 and XL is safety off at pectoral index, (as Maple Leaf Actual pointed out), then back on before moving to the holster, moving, or coming to the ready position. If I come to the ready, holster, or my feet move/shift position behind cover the safety comes on before I move. You can trip and fall, smash into cover, other objects, or other soldiers I would not want to risk an ND because it was just easier to leave the weapon on fire. Doing so is just lazy and negligent handling. Negligence will lead to bad shit, it has, does and always will. Not trying to be condescending but really think this through, bad habits take forever to undo, you can't take a round back once it has been fired and lastly if you have poor habits with the pistol there is a possibility it will bleed over to your carbine work. I try to keep everything the same, less to F' up.