I know when your cleaning a lot of records or if you use a bit too much cleaning solution the strips do get a little saturated and will start to smear, that's an easy fix. But, it sounds like your wand isn't sealed, it sounds like there might be a small crack somewhere or the plastic is just getting old and brittle.
I replaced mine with a
Delrin wand a while back and it's been fantastic, I'd recommend thinking about going this route if you end up replacing your wand over going with another plastic one.
I'd say it's definitely one or the other, if not both.