It's easy to tell how much oil is needed, just put the shock vertically in a vise to old it and unscrew the filler plug on the front side. Fill until oil overflows. This leaves a small airspace at the top.
I also have used motorcycle fork oils. Other non-detergent oils might work fine too. One problem with fork oil is that there doesn't seem to be much consistency of viscosity ratings from one brand to the next. One brand's 20W fork oil might feel about the same as another brand's 40W and a third brand might just call it "heavy duty" and not list a viscosity rating at all.
I think the original Armstrong oil is about a 15 or 20W. Moss sells something that seems about this weight, just from feeling it because it also doesn't show a viscosity rating on the label. Some folks like to use a moderately heavier oil, say 30 or 40W (or a blend to make approx. 35W), for stiffer shock action. In extreme cases I've heard of using 50W racing motor oil in the shocks, but that sounds awfully heavy to me.
Another way to modify the shock's performance is to replace the dampening valves. On TR4 (and I presume on later TRs since they are close to the same weight), one common mod in the past was to use valves taken from lever shocks on the front of MGBs. These uprate the shocks approx. 25%, I've heard (haven't done it yet, myself). There is an article somewhere on the internet about other possible modifications to the shocks.
While its still in the vise, operate the shock lever by hand a number of times to be sure all the air is out of the valve mechanism. It should feel sort of "gravelly" at first, then get smoother and smoother operating. Finally, the shock should give nice even resistance both directions, smoothly and without any roughness in the stroke.
If you feel any uneveness or roughness, the shock might need to be rebuilt. You can try carefully removing the valve mechanism at the bottom (the large nut) and cleaning it before sending it off for a rebuild, though. It could just be something trapped in the valve that's preventing it from operating. Be very careful, though. There are springs in there and it want to fly apart as you remove it. Also note the parts sequence, it all has to be reassembled in a specific order for the valve to work properly, too.