I let the body go into the ms, I try to keep the dial out of it. let it sit a while, then I drain it and stand it up after a while I blow it off with about 5-10psi.. just to get the remaining fluid to release from the parts, you could swing your arm with the probe hanging down it would do the same.
Mineral spirits has very light lubing qualities, and it degunks the mechanism. Its worth a try. Most dials are not paper, but I just did a federal 10ths last night and took the crystal (plastic ) off because I wanted to rub the lens with toothpaste to clean it up. and I think it was paper. I also wanted to see if I could get the arm off to see about straightening the shaft, it's wobbling. BTW toothpaste on a lens will help clean it pretty well. It took most of the yellow out, and almost took the scratches out. This one may go out due to the shaft being bent, but hey I only paid $15 for it.
edit: let me add I have done quite a few other indicators, Mitituyo, federal, bestest... all get this treatment when I get them if they exhibit any indication of stickiness.