Hi Aaron
I don't have one of these myself but hopefully someone who does may be able to advise on the specifics. Ideally they would be able to give you a copy of the servicing manual in PDF format.
There are likely to be several causes contributing to the backlash you are seeing. As I think you are suspecting, the leadscrew itself needs to be adjusted so that it doesn't move left to right under load – this would presumably be where the handwheel is fastened. Hopefully somebody will confirm that. The gib screws on the saddle (and also the cross and top slides) need to be adjusted so that the saddle remains square to the bed (ie it's not too loose) and finally, the leadscrew and half nuts themselves are bound to have some wear. You can't do much about the latter short of replacing one or both of them but it's probably not an issue unless you plan something really precise and critical.
For facing off, you should think about tightening up the saddle lock so that the saddle movement along the bed is stiff or even fixed, For lengthways turning, you mainly need to take care not to forget about that backlash if you are threadcutting. Generally I think you'll find most people don't rely on the dials much, particularly for lengthways measurements and instead use calipers or digital readouts to measure what is actually happening. Moving the saddle when turning would normally be done with the apron handle, not the leadscrew, so in most cases the backlash wouldn't be a problem.
A good starting point would be to adjust all the gib strips and any other sources of backlash. Looks as if you will be able to do this as you reassemble your machine. Giving it a good clean and keeping it clean and lubricated is a good idea obviously.
Hope you have fun with your machine! You do realise of course that you should waste no time acquiring milling machine, drill, welding gear, hand tools etc etc – it's a life's journey!
Murray