In the event the above suggestions don't cure things, you may be forced to have a peek inside. Lots of cassette decks use an elastic band to connect the motor to the capstan (I am sure the manufacturers would rather I called it a drive belt or constant-velocity-variable-speed-diffidoodle, but its basically an elastic band). You should be able to get a replacement from the manufacturer a lot more cheaply than putting the deck in to be repaired and replacing them is very easy. Whether this is a likely cause depends on how you trade off "pretty new" against "which I use a lot with students"!