I would doubt the fluid is causing the problem, but it's certainly not a bad idea to flush and refill the system with fresh fluid. Make some effort to drain the old fluid from the steering rack and suction chamber of the pump before refilling the system.
Also whilst you've got the system drained it would be prudent to check the flow control valve in the pump.
You'll need to remove the union bolt and pressure feed pipe from the top of the pump
Then remove the pressure port union, flow control valve and compression spring
Then check the condition of the valve and spring. If the valve doesn't drop into the hole very easily, then soak a scouring pad or scotch pad in some ATF and give both the valve and valve hole a light polish and see if it's any better.
If you don't find a problem and considering no play can be found in any suspension/steering components then the problem may possibly be in the rack. There might be a blockage or some wear around the fluid return port at the base of the pinion. The dark/dirty appearance of the fluid is something to be a little concerned about, the fluid isn't subject to any extreme temperatures so won't burn like ATF would in an auto transmission. The most likely cause for the fluid discolouration would be the deterioration of the inner walls of PS rubber hoses, the seals in the pump or rack or possibly the dreaded teflon rings on the pinion shaft.