I would need more info to decide if you're safe or not. What size and how many NiMH are in the radio and will it accept the same # of Alkaline batteries? As long as your lightly loaded supply doesn't exceed that voltage by much, the radio is probably safe. Then it comes down to the safe charging rate for the batteries and what you're willing to do with your adapter. Tweaking the adapter for a good charging rate could be as easy as inserting 1 or 2 forward biased diodes in series. 1N4001 series diodes are very common and fine if you keep the current < 1Amp. If you need 2 diode drops for the right charge rate, you can use one of those small cylindrical bridge rectifiers that I think Radio Shack still sells. Just use the "-" towards the supply "+" and the bridge "+" to the radio "+" input, while isolating the "AC" leads. The current rating would be twice that of the bridge because you would be using it as 2 pairs of series diodes in parallel. I would just try it and make sure that the batteries don't get too warm during charging.
Joe
