Yes, clock wise to remove play. Do it with the jack stands under the front suspension. This keeps the front end the same as it is when on the road. When winding the steering back and forth aross the centre position you shouldn't feel a stiff spot at all. The screw is coarse thread so a few degress of turn makes a big difference. This type of steering box normally has some play at the steering wheel. About an inch each side when new and about 2 inches each side on a worn box not unusual. Doesn't normally affect driving. Adjusting this play out and getting the stiff spot is bad and will increase the wear rate as well as giving you the wandering you are getting. It should steer straight, hands off, on a good flat road. If it veers of to one side or the other, check out all of the front suspension joints, alignment or for slightly sticking calipers.
If you took the Pitman arm off the box when you refurbed it last time, check that the arm went back on in the right place. With the nut off you should be able to see a mark on the arm and the shaft. These must match up and the arm point straight ahead when adjustment, and tracking, is done. It also ensures that the internal steering gears are correctly centred. i.e. at their slackest, and the position that they should be when adjusted. Ignore the steering wheel position, that might have been put back on a spline or two out. Correct that last.

If the adjuster screw is level with or below the lock nut, the box is badly worn. A couple of threads showing is OK.

Good luck and I hope your driving lessons are taking place off public roads and go safely!