After spending a few hours across several evenings, I have the ink fountain from our Thompson press cleaned off and fully disassembled.
It took several treatments with the paint stripper to get everything clean. There was no way to distinguish dried ink from paint so I stripped everything clean and will repaint.
The adjustment screws eventually came out after the paint stripper had softened the ink on them and they now operate fairly smoothly, considering that I have not oiled them (I want to avoid oil until I have repainted the parts).
My next step will be to repaint the parts. Only the lid and part of the main body are painted; the rest is bare metal protected from rust by a film of oil. The rest of the press seems to be finished with a semi-gloss black so I will be using that as well. I will probably also use some contrasting paint to bring out the raised lettering on the lid.