how steady over time are right ascension and declination values? I understand that astronomers use right ascension and declination for the position of stars. However both these values strike me as very anthropocentric and, more importantly, unreliable over time as I understand that the Earth wobbles a bit over time so the celestial equator shifts and I guess Sun's rotation around the galaxy and Milky Way's own trajectory should play a role in depreciating the accuracy of those values over time.

So my questions are:

[1] how steady are right ascension and declination values for distant stars over time?

[2] is there a more stable coordinate system or do we lack an absolutely fixed frame of reference?
The values are more steady for objects close to Ecliptic poles, but in genenral, they change by a few arcseconds every year.

Yes, we lack an absolutely fixed frame of reference.