Moving fast doesn't screw radio signals up. Not unless you move at relativistic speeds in which case all it does is change the speed (the frequency and the data rate) of the signal. Not "jumble it out".

But 6 km/s while it sewsmc fast for us at ground level, it pitifullly slow when comparing to light speed. the relativistic Lorentz transformation effect aat such relative speeds is so tiny as to be completely ignorable.

And there is no atmosphere in space, either, so moving fast won't shake you around like a plane flying too fast beyond it's structural limits and being destroyed by air turbulences. There is no "wall of air" in front of you to constantly apply friction and slow you down: there is (next to) nothing! Basically, ideal conditions for moving around.

It is not the engine's power and speed that gives off that 6 km/s. It is the slow ACCUMULATION of speed by the engines. Forget Hollywood sci-fi when they nearly instantly reachh full speed when they lit the engines, then magicallly slow down to a stop when they turn the engines off. Things don't work like that in reality. At all. Think more like this: your spaceship has a speed vector. Each time unit, you move by that speed vector. It doesn't matter if you rotate where your ship is pointing at, it moves in the same direction of that vector. That is called inertia: things tend to keep on moving the same way unless a force is applied to them to counteract that. Now, you have engines, but all they do is, each time unit, add a TINY speed vector. Say, after accelerating for one hour, you are now moving 5 km/s from bottom to top. You could turn the ship sideways to turn to the "right" and lit your engines for say another hour. Then you'd be moving the same 5 km/s from bottom to top PLUS 5 km/s from left to right, thus now your speed vector is about 7.4 km/s going "top and right".

It takes a long time to accelerate something to 5 km/s. For comparison the fastest hypersonic jet known movves at Mach 6.72 = 4520 mph = just about 2 km/s. And that is with a jet with a superbly monstrous and HEAVY engine, that can use the abundant "thick" air it travels through as for it's oxygen for nburning it's fuel much hotter. Not the measly "built to work in space" engine of a little very fragile satellite. These two things can't even be compared.

Presumably, DART's propulsion systems were turned off for its very final segment, letting DART final closing in "sail through" mode, without any engines vibrations in other to get the best images. No atmosphere, means no vibrations and no friction. And very clear images, too. Once you turn the engines off, you just keep on moving inertially at the same speed.

From the point of view of DART, once itS' engines are off, it is immobile and it is Dimorphos that is closing towards it at 6 km/s. Not the other way around.

Because Special Relativity, ya know.