Start with the easy option.
Take one tuner apart, noting the order in which it goes back together. Clean up any dirty parts (alcohol and a toothbrush is as good a way as any, unless there's substantial corrosion, which seems unlikely). Let dry and reassemble. A tiny smear of ordinary lubricating oil on the screw thread and nowhere else (you don't want oil on the wood, or on the parts which are friction!).
Now reattach the string, pulling it all the way through the tuner hole before you start tightening, so that when it's up to tune there are only two or three turns round the post. Lock the end of the string under the first wind round the post. Pluck away on that string, tightening it as the string stretches. As these are not new strings, it should settle down within a couple of hours.
If that string now holds its tune, you can do the same job on the others. No searching out for washers, no expense, just 20 minutes work. My prediction is that this will fix your problem unless your strings are old and dead. If it doesn't work, all you've lost is a little time.