Is there any way to gauge this? I'm really sort of worried that I'm destroying my cuttings.

The general rule I have heard is that if there is condensation inside the cups  you are good, no pooling of water in the cup, and something around 70% humidity. I keep a small reptile terrarium hygrometer inside the bin I put my cups in. 

If you can squeeze even a drop of water out of it.. it's too moist.  If fact, if you completely soak it and squeeze out every last drop, it still may be too moist.. but that's how what I do it.

using a bag? or a box?

Yeah. I had it way too wet. I found one of my cuttings rotting, and I'm worried I've doomed them all. Well, there's my first experiment with figs done. I'm hoping that I may get at least one to survive from this mess. I'm going to feel like utter garbage if this is a wash, because someone was nice enough to gift me these.

Root in a paper towel in a ziplock bag on top of your fridge (or in another warm but dark place) in parallel. Make sure the paper towel is damp, not wet.
Dark-> roost go first, light-> leaves come out.

I use a fine spray bottle to hydrate the moss i.e. throw in a handful of dry moss in a 1 gallon bag, spray a few times, throw in another handful, spray, and so on.  If anything, I aim to underhydrate at the beginning.  Then after the cuttings have been on the moss for a day or two I might spray a couple more times if it doesn't seem damp enough.  You want it to be damp but not wet.  I have heard of people putting in a dry paper towel if they feel they have added a bit too much water.  After it has absorbed moisture it can be discarded.

My usual line.....try what works best for you.  Sounds like the spaghnum moss route isn't working for you so try another method.  Root riot cubes or a mix of potting soil/perlite in plastic cups.  I didn't do very well using the spaghnum moss routine but did really well using a perlite/potting soil mix.