From the symptoms, looks like the azimuth of your play head needs calibtation but that is only if your deck's pinch roller is in good shape. Not sure what exactly you mean by tape-eating but assume you hear a squeezing sound when the deck gets past the leader and the tape hits the pinch roller. When you hear the squeeze, you check the tape and it appears folded or crushed at the lower or upper end? If this is the case, then its your pinch roller. When the pinch roller wears out, it moves the tape upwards or downwards (depending on the wear) on the capstan, resulting in muffled sound and chewing. Also, if the pinch roller is worn, it may move the tape running past it, slightly off position on the capstan which will result in muffled sound. Does this happen for all cassettes or only specific ones? Try out thicket tape like a C60 and compare it to thinner tape like a C90. The thinner the tape, the more pronounced the roller related issues.
Make a start by properly cleaning the pinch roller and capstan. The pinch roller should not appear smooth but don't try to rub it and roughen it if its smooth. If its smooth, it would mean that its worn and needs a change.
If its not the pinch roller, then the azimuth would need calibration.
