The shutter sound is inevidable, but if you can deal with a beep, the shutter is no big deal.
The noise that is making your subjects jump and is probably not the shutter. Once the shutter is pressed, save for blinking, few things will ruin the picture. People can't really react that quick, might ruin the second picture but not the first.
The noise which is probably setting them off is your autofocus motor! That noise will. Simple solution, manual focus your lenses! No need for a new camera...