Do NOT grease linear rods. Clean them thoroughly (I use IPA, take care not to get it inside the bearing), then very very thinly coat them with light oil, then gently wipe it off. Those bearings are (should be :)) lubricated from the factory. If you feel like they are dry, then the only solution is to frequently apply small amounts of light oil (possibly push on them a bit to make the bearing inside them roll rather than slide), but that should only be last resort. If in doubt, just wipe them with a clean rag, and beware of any strands or dust od debris getting inside those bearings.
I think there are more factors than one to what’s happening in there. I found that making the belts tighter might improve the problem for a moment, only for it to return later. Sometimes doing nothing and just calibrating is enough. I’m pretty sure that it’s the resonance compensation mechanism trying to cope with play somewhere in the system that is not actually a resonance, and mostly failing.
I found play in my toolhead (between toolhead and the bearings), like described here Extruder has play in Y-Axis and will try fixing it tomorrow. Whether that is a complete fix or not I don’t know, but this is the only play I can see and it makes sense why I didn’t notice it when manually moving the toolhead - if you have tension on the belts, then it masks this problem, but the motor actually tensioning the belts in opposite directions will make the toolhead warp because of the play. At least I hope that’s the right explanation 
To create a test cube, create a new project, right click on the bed in the Prepare tab and then Add Primitive → Cube. Resize to 50x50x50 and print. But it’s enough to just print anything and use Rectilinear infill, it should show on any print.
I’m convinced I’ve had this problem for quite a while and just didn’t notice. Over time I got used to some things getting very slightly worse, and since I mostly use Gyroid infill and filaments that hide errors, this went unnoticed. Maybe something in recent firmwares made this worse…