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I hear a trampoline spring type noise when I apply light pressure when taking off slowly. It sounds really weird, I have no idea what it is. Like if you were bouncing on a trampoline and the noises the springs make that's what it sounds like. 93 f150 4.9 4x2
I would look at drivetrain parts (I.e. wheel bearing, u-joints, breaks, ball joints). Do these truck use a speedo cable? Could make noise inside the protective sleeve
U-joints can start making a noise like that when they start going bad. It will progressively get worse if that's what it is. See if they have any play in them and if they have a zerk fitting shoot some grease in them and see if it helps.
I've got a bad motor mount and an iffy trans. mount on my truck (1990 f150 4.9 5 speed) and it makes a noise similar to what you're describing. You might want to check that as well