Before getting through your post, my first thought was to return the drywall. If it by mislabeled you mean the product say on it self that it is 1/2, you could try and get your money back from the store, but processing it all might have made it difficult to do so. If you mean it might have been in front of the wrong sign at the hardware store ...
My second though was what was does your basement look like? Perhaps there is a corner you could switch from 1/2 to 3/8, even if it meant buying a couple sheets of 1/2. Or failing that some other transition point where it would be less noticable (or easier to mud/conceal), say behind a floor-ceiling cabinet or a stack laundry unit.
My third thought was shimming. I would suggest something other than paint stirrers. This is a bit of a kludge, but if it is acceptable to you ... there are worse things lurking behind drywall. If you have a table saw, I would rip 2-by material down to 1/8 thick and do full furring.
My fourth thought was a mud transition, but that would have to be a long tapered transition. I know that it would be beyond my skill to do so in a manner that, once painted, would be acceptable to look at.
My fifth thought was, how much drywall are we talking about here; 2 sheets or 34 sheets? You still might be better off in terms of your time and cost to repurchase the correct drywall and mark this up as a lesson that stuff at the hardware store is often put in (or dumped by other customers) in the wrong place. Or maybe, if cheaper/easier rip of the remaining 1/2 and replace it with 3/8?