The simplest and quickest way to check for headset play is to lift the front end about a foot and drop it onto the floor. A loose headset will give a very characteristic "knock" or "clunk", and a tight headset won't.

The other test is to apply the front brake and rock the bike forward and back while looking and/or feeling for movement at the lower bearing.

The important indicator of a loose headset is a slight click or clunk when applying the front brake, as the fork pushes back in the bearing.

It's normal to feel the bearing somewhat when a headset is tight, but there shouldn't be any binding. If you can't get it adjusted to find a balance between no play, and no drag, there may be misalignment someplace, or if play is very difficult to resolve, sometimes the headset is OK, but one part is loose on the fork or in the head tube, ie. a crown race loose on the fork.

So it you're struggling with the adjustment, and it's not just you take it apart and see if you can spot the issue (other than adjustment). Among others I've seen retainered balls upside down, cups loose in the lower head tube, integrated headsets not seated properly, or mismatched for the frame, upper centering cones missing or damaged, and so on...

First try adjusting, then do a thorough diagnosis to find why you can't.