What you’re twiddling with are calibration potentiometers and yes, very sensitive. I couldn’t tell you how many times I’ve thought “that’s it, that’s calibrated!” and then simply removing the screwdriver knocks it out again. Most turntables will have these pots, but a lot require partial disassembly to access. I’ve worked on a lot of turntables (generally older stuff) and I’ve figured most of it out the hard way (so I’ve made plenty of mistakes).
A couple of things you might want to consider before calibration is the age of the unit, the belt (if it uses one) and whether it’s been sitting unused for any period of time.
For example if I’m pulling a 40+ year old turntable from someones shed; I’m going to be lubricating both the motor and main bearing (possibly requiring a total strip/clean beforehand) and most likely fitting a new belt. A drop of oil isn’t a bad idea purely from a maintenance perspective though, it’s not just a restoration thing. After doing this, I would want to let that lubrication run-in for a good 30 minutes before attempting calibration. Other things like aging components can influence power delivery to the motor. Something to be aware of, but often can be overcome with speed calibration.
One caveat with smartphone RPM apps; while the weight of your average smartphone isn’t that far off the weight of a 180g vinyl record, plonking it down randomly will result in uneven weight distribution and this could influence the speed (particularly wow/flutter) while you’re trying to calibrate it. If using such apps, I’d recommend sitting your phone above the spindle by using a 45 adapter or better yet something like a small roll of sticky tape. This will allow you to center the weight more accurately.
Smartphone RPM apps are surprisingly accurate but my preferred method is to use a stroboscope (printable, just make sure you’re printing to accurate 100% scale and have picked 50/60hz depending on region) and wow & flutter meter.
A wow & flutter meter is not the kind of thing that most people are going to own (myself included), but software alternatives exist! Granted, they require you to have some vinyl containing test tones and some sort of basic audio interface for your computer. If you dig for secondhand vinyl, you might want to start grabbing those stereo/hi-fi setup/test records as they will usually contain some (can be useful for identifying tracking issues, too).
The software I’ve used is really aimed towards servicing tape decks, but I’ve found both ‘WFGUI’ and ‘Real-time frequency counter’ to be very useful: http://www.ant-audio.co.uk/index.php?cat=post&qry=library
They may be intentionally set a little fast as-new. This could be to compensate for expected slowdown associated with aging of the unit. For example; running 2% fast from new might be less noticeable than running 5% slow after a couple of years. This is just a guess, though.