Well, if it was a model that was more common, and if I had a schematic diagram for it, I might be better able to comment. Without that, I'm only guessing.
With most AC products today, the cords are polarized except for those appliances which are "double-insulated." AC polarity is not to be confused with DC polarity because AC itself isn't really polarized like DC is. The reason AC polarity is important is because most appliance chassis are tied to the neutral lug. The other lug (hot) can kill you. For this reason, you never want to reverse the polarity on devices that require polarized cables. I have seen polarized cord ends shaved down so that they can fit into sockets that were not meant for them. Going back to the neutral lug, that (and everything else around us) is to some extent, connected to it. That means that if you touch a hot cable, the current would want to go through you, through the carpet, floor, etc until it gets back to ground (or neutral circuit). It is for this reason that the cords are polarized, otherwise if reversed, the chassis (or cabinet of your tool, refrigerator, etc) would be hot and touching it would be a huge problem.
On a home I just remodeled, there were many receptacles and a couple of light fixtures that had the hot and neutral cables reversed. I found out the hard way when I went to use my skillsaw connected to that receptacle. I almost got electrocuted simply by handling the metal case of the saw. Then there was the fan. When I went to change a light bulb, I carried a no contact current sensor to verify that the electricity was off before removing the bulb. Except when I stepped on the ladder and got closer to the fan, the sensor went off like crazy from all over. Evidently (and I confirmed this), every metal part of the fan was "hot." The most frustrating part was when I went to remove the receptacle to switch it back to the proper polarity, I found that the wire was actually the proper color, meaning that whoever "me-wanna-be-electrician" idiot did the work didn't even connect the wires properly. This is hard to do since black should always be connected to black and white should always be connected to white (unless routing to a switch which confuses some people). But I digress.
On the boomboxes most of us are interested in, the AC connection isn't really that important with regards to polarity and that is because internally, the transformer is acting like an isolation transformer. The current goes in one conductor, goes through the transformer winding, and returns through the other conductor to the receptacle. Since no part of the boombox including chassis, should be connected in any way to the transformer primary (the mains side), it should be "isolated" and therefore not an issue. Remember that AC has no real polarity and it doesn't really care one way or the other, will still work -- it isn't until it's rectified by the diodes that it becomes (+) and (-). The truth is that even if the transformer secondary was 100v and not 15v, if you were to accidentally touch ONE of the secondary conductors, you would not receive a shock unless you were somehow touching both secondary conductors. That's because the secondary is isolated from the primary so if you were to hold one conductor, (your hands, feet, etc are only technically connected to neutral of primary side and not the secondary side) there is no return path to the other secondary conductor. This is the same reason why birds can perch atop those high voltage cables without getting electrocuted. This is not the case if you were to touch a conductor on the primary side. One conductor is neutral and we are already connected to the neutral because every house has either a ground rod, or has the rebar grid within the house slab (although the wood, carpet, shoes, socks are all pretty high resistance). So touching the neutral does nothing since if there was any current there, it would simply pass through the conductor easier than it does our body. However if you touch the hot connection, the current will want to pass through YOU to find the path back to neutral (or ground) as resistance through you is less than air.
Ordinarily, the secondary of the transformer should not have any connection to the primary side. However, it is possible for the transformer windings to have some leakage if the insulation begins to break down, or even in some cases, by design, with special capacitors, leakage increases to unacceptable levels and you might observe the issues you speak of. Even electrical cord insulation, although a good insulator and conducts very little but very little is not none. So YOU are always at some level of ground potential and if your radio has some hot potential to it, then you might be able to "ground" your radio through you. The fact that this radio has a built in cord makes me wonder how it is connected internally, but the best solution for you is to simply replace the cord with a proper polarized cord. If your old cord is leaking some current, simply replacing it without regards to polarity might already fix, reduce or minimize your issue but if you replace it with a proper polarity cord, should eliminate it.
If you are really anal and wish to eliminate any chance of AC isolation issues, you can get something called an isolation transformer. Hospitals use them a lot. It's really just a transformer where the primary and secondary windings are matching. So 120v in = 120v out. The difference is that since the windings are isolated, you won't get a shock if you were to come in contact with any single conductor since there is no return path to the primary ground. Only if you were to touch both conductors would you be in danger of getting shock or electrocution.
I take it you've been to my house. 