I don't know what other people do, do I wipe them out first with household paper towels, then watch paper, which is just lint free paper, and finally I press some old rodico into it to remove any fibers from the papers.
Other than for applying grease to the barrels I usually just use me smallest oiler for everything else on watches.
To get an idea if you are applying too much oil assemble and oil a watch, wear it for a week and then strip it down.
If you see oil has crept all over the place you have used too much and use less next time.
I never realised I was over oiling until I stripped down a watch I had previously serviced when first starting.