The cassette tape is fed past the record/playback head at a constant speed by a rubber 'pinch' roller that pushes the tape against a revolving metal spindle. The surfaces can get dirty over time and a bit slippery. This allows the take-up spool to pull the tape through at a faster speed which will tend to make the sound 'wobbly' or a bit towards the 'chipmunk' sound you mentioned.
It might help to clean the surface of the rubber pinch roller and the spindle to remove any surface glazing or marks. I'd use a cotton bud dipped in isopropyl alcohol which I have for cleaning vinyl LPs and stylii, but I am sure you can find something else that will help - maybe just the corner of a tissue soaked in warm water.
No promises that this will solve your problem but it is something I used to come across when I used cassette tapes. If this was the problem, I would have expected it to have affected the recoding as well so I have my doubts, but it is a worthwhile bit of maintenance anyway.
Another possibility is that the cassette tape you used had not been wound through for a long time and created extra drag so that the actual recording was too slow. Now you've used the tape to record the music, it runs more freely so it plays back at the proper speed, making the playback sound 'chipmunk'.
If that is the case, you will have to re-record the vinyl albums on cassettes that have been run-through recently.
Just a couple of ideas. Hope you solve the problem.