Firstly have someone hold the steering wheel steady while you're doing the tyre shaking test. Sometimes the steering wheel shakes a bit while doing the test and I've been tricked many times into thinking there's some play at the rack.

Its possible that your inner tie rods (right and left) joints are worn. These are bolted to the steering rack itself. You'd have to take out the steering boots to be able to see them. With the boots out (but still hanging on the inner tie rod shafts) shake the tyres again to see the loose joints.

And if you see oil dripping from the boots then the rack assembly itself should be replaced or rebuilt. If the boots are dry only the inner tie rods should be replaced.


Regards,
Amarin.