The hyperglide cassette lockring only holds the cassette against the freehub body at the ends of the splines. It should not effect the smoothness of the hub bearings, unless you tightened it so much that you distorted the freehub body.

You should remove the cassette from the freehub and then properly adjust the hub.

First adjust the axle so no motion is felt perpendicular to the long axis of the axle and it still turns smoothly. Rotate the axle slightly and check axle adjustment and continue to check in several rotational angles. Ensure that the locknuts are snugged tightly against the cones (assuming ball bearings with cup and cone hubs) and re-check your adjustment. If you cannot adjust the bearings properly, it is time to tear the hub down and replace worn-out components. Always replace the bearings.

Now try installing the cassette. Tighten cassette lockring fairly tight, certainly more than finger tight, I usually use a long handle (14") adjustable wrench and torque pretty tight.

Also, almost all freewheels/freehubs oscillate or move slightly in operation. I remember hearing somewhere that this was intentional, but I cannot recall why...

Good luck.