Most Pillar drill presses are the same only the sizes of bits differ.
The causes of ‘runout’ are:-
· A bent drill spindle.
· The chuck is not concentric with the shaft.
· The shaft bearings are worn.
· The spindle is worn.
· The quill (the bit that moves up and down carrying the spindle) is:
o It is worn
o Its sliding bearings are worn.
o It can rotate slightly and the spindle is not concentric.
· The spindle is loose in the quill. The spindle nuts need adjusting.
The only way to find out is to slowly strip the thing down checking, shaking and measuring as you go until you find the cause(s) of the problem. There may be more than one problem.
You ask about the nuts which hold things together. If you look down on the drill from above (or at a drill) you will see it rotates clockwise. The nuts will be made to tighten up in that direction of direction.
You say it has spent most of its life drilling wood I would suspect the bearings. Wood dust and oil make a good abrasive.
By the way there is another type of drill press. Here the spindle does not move up-down but the table does.
Regards
Dick