It is hard to say, but I would look at the diode first (test and/or just replace it). And put that AC power thing someplace where you will never grab it again. I have seen ICs get hot but still work when you fix the problem (sometimes, not always)
I own four "one-spots" now. I see them for $10 online & stuff all the time. I just bought another 9v DC adapter on Amazon which comes with six output cable for just $10 shipped. The only drawback is that it is center+. But that also give you a turnaround adapter. $10 shipped.
I don't know why anyone would use a battery anymore (except for cases like yours of course). I am lucky, I have burnt things out before with my one-spot, but not because I used it wrong (because there was a fault in my build). I figure I am going to use a power unit anyway at some point, so I might as well use one in testing.
I tested the diodes which were fine and replaced them anyways. I think i'll just order a new IC and then a couple of electrolytics just in case and go from there. I had just ordered a new power supply but thanks for the advice on the one spot.