DC Power Supply Safe Ranges
It would depend on the radio itself. 3.7v might not seem like alot, but it's a 25% difference, but you never know how tolerant the circuit inside would be to higher voltages.

Additionally, your adaptor might be a cheaply regulated one. It might only provide the rated voltage, at near the rate current draw.

If you can open it up, you could see if it has an internal regulator, but a 3v circuit on a cheap device, it would probably only have a diode or resistor.

Is the new adaptor also center pin Negative? The standard is normally center pin positive (have not seen any center pin negative power supplies in a decade, but they are not rare either). Having the wrong polarity would definitely cause problems.

As you stated, yes, the current rating is okay, because that is it's maximum output capacity. The radio will only draw as much as it needs/was designed for, and undoubtedly, it will draw less than the 300ma it lists (design tolerances).

(Personally, I have and would use such a minimally different wall wart. But it's not recommended, and only done it knowing that it could possibly damage the device. If you arn't willing to replace it if it breaks, don't do it).