"What could be the reason that when I connect my WM-D6C with the adapter, it winds the tape in reverse, pulling the tape in instead of playing it in the correct direction?"
Not sure, but it maybe you have the wrong polarity; in which case you have already damaged the unit. The polarity for Sony D6C is centre negative.
Get rid of the adaptor and use re-chargeable NiMh batteries only. As @TooCooL4 says, unfortunately you may already have killed the CX20084 motor drive chip. Sony WM-D6C and DD Series Walkmans - WARNING! Read this before plugging in any DC adaptor!