There are two screws, one at the back of the headstock and one underneath. The tailstock clamp screw also locks the arrangement, so it is a bit of a fiddle. I made a brass gib for mine to make it easier to get free adjustment, but for now I suggest you go with loosening both screws then retightening to get as near to smooth but shake-free movement as you can get. Make sure no paint has wicked onto the sliding surfaces. Lock the tailstock barrel in place using normal force.

A basic alignment is just to have a centre in head and tailtock and align them using short sight or a loupe magnifier. If both centres are sharp and accurate this gets you surprisingly close.

Interposing a steel rule will highlight any misalignment very well.

A practical test is simply if turning between centres gives you a bar of constant diameter – you can turn away a lot of bar chasing (and overshooting) the last fraction of a thou this way.

Best practice is a dial test indicator held in a chuck and used on the tailstock outside surface to check it reads the same at both sides (and hopefully up and down).

The 'practical' route, as used by LBSC and others is a long 'test bar' between centres with the centre turned down by 50 thou or so to give you a long 'cotton reel' with flanges at the ends. Turn the 'cotton reel', and then skim the tailstock flange lightly, reverse the bar and skim the other end at exactly at the same setting. Use an indicator held in the toolpost to take readings from the flanges and when they both indicate the same the bar is parallel.

Keep the bar safe or future use.

I don't do this as I'm the sort of person who ends up using the bar to make something…

Neil