This response is in the camp of the guitar having the problem. 
just a couple easy checks I haven’t seen recommended:
- Verify the nut securing the tuner is tight. This is the threaded nut, not the guitar fretboard nut. I had one of those loosen and had all kinds of odd tuning instabilities. Tightened up and no problems since. Don’t let the wrench slip or you can round the nut corners.
- Make sure the string is properly seated in the guitar nut. If it is out the action will be high among other things. If it is out of the slot, loosen the string before slipping it back in so you don’t chip the edge of the slot. Check the bridge end as well for the same thing.