Before adjusting any of the box adjusters , you want to check all front end components and adjust the front wheel bearings.
There are two box adjustments . One is the block pressure adjustment [ the top screw/lock assembly] and the other is the end float of the worm gear assembly.
The worm end float has to be checked before setting the block pressure , as attempting to take play out of a box with block pressure only when one has worm gear end float is likened to adjusting your wheel bearings after you set your chassis toe alignment ..
The best/recommended way to do these are with the box out , as it can then be set to proper torque load specs , but if one pays attention , you can get it in car..
The first spec to look for is 25mm max movement at SW circumference
before output shaft [ pitman arm] starts to rotate.
To eliminate front end component wear , it is easiest if you have someone turn SW back/forth in decreasing amounts til you relay to him that there is no longer any output shaft movement .. this will be the actual box play and should not excede the 25mm [ 1'] at SW circumference. If this is in spec. , your problem is not in the box, but rather the front end components or chassis alignment. Once that is established and you find the 25mm is exceded and is the results of the box , you now want to check for worm gear end float. You can install a centering tool in the hole of the hex head plug on top of box [ to the left of block presure adjuster] and set the tool into the central detent of the worm assembly. This locks it and then you can see if there is end play in the shaft . The end float is spec .00mm - .01, so you can see they do not want any, but they do want the worm to still be free to turn .Again , those are bench specs.. you just want to see if the shaft has visible end play..
The adjustment on this is done by tightening the ring with a pin wrench and lock nut , which sets the end load and float on the worm.
If there is no discernable end float, you can continue on to adjusting the block pressure . This adjuster increases pressure on the block when it is screwed out. Again, this can be exactly done on a bench to spec of 1.75mkg - 2.25 mkg torque force required to turn the shaft throughtout it full arc , with the pressure dropping as the movement goes through the center position. That is what allows the SW to automatically return to center after a r/l turn. Again , box out/bench talk... ..for in car, you can just make 1/8" tightening increments ..
I recommend taking the car for a trial run after each incremental adjustment and try both r/l turns with attention paid to the SW returning to center . If you go to far , you will notice the cars ability to return deminish as pressure block adjustment becomes too high.
With the box on a bench,one can set these specs w/torque readings
to perfection, but .. seldom does anyone do that [maybe a few]
Lastly, and not pertaining to the SW play .. your condition of having to put several steering inputs at speed can be a rear end suspension problem.. these are also often overlooked/ not considered , but they do steer the car on straight away travel .. suspect here are both upper/lower hanger bushings and improper axle cross strut axle centering adjustment/rubber bushings... test checking for excessive torque steer will be an indication of a problem back there....
I am sure there will be other replies..clearing the mud up....