
If the stains are already there you should find they move a little at a time- You may have to clean them every day. I find going against the fur helps and gets right to the root. Afterwards try a little vaseline where the stains appear. This makes any tears run off rather than stain. You will remove this the next day when cleaning eye area again.
Also worth checking eyelids that all hairs are growing in the right direction and not irritating the eye.
IMO tears are ok during teething but I would want to find out why if it continued past teething