Power Supply 4.2VDC broken. Freedom of choosing new one
The power supply in question is for charging a lithium-ion battery. These sorts of batteries have special requirements for charging in which a constant current (1.2A) is applied until the battery reaches some voltage limit (4.2V) and will turn off once the current has decreased to some limit (unknown, but probably 10-100 mA).

It is unclear if this charging circuitry is inside the battery-operated-device or the adapter. For the following analysis, I'll assume that it's in the power adapter, which seems likely.

The adapters specified in the question are not battery chargers and also do not limit the output to precisely 4.2V, so would be dangerous to use in this application. An adapter sold as a lithium-ion charger would be appropriate.

The battery could be damaged by a battery charger that has a HIGHER current limit, but it would be safe to use a lithium-ion charger with a LOWER current limit. The lower current limit (such as 800 mA or 1000 mA) would charge the battery more slowly, but would be safe and actually allow the battery to have a longer life.

Additionally, you need to be aware of the physical shape of the charger port in order to ensure mechanical compatibility. It may be required to retrofit the new charger with the plug of the old charger (using crimp connectors/soldering/etc....). Be sure that the two wires are not reversed (as this would damage the battery, perhaps causing it to catch of fire). The old adapter may indicate which pin is (+) and which is (-). A voltmeter could be used on the new charger in order to determine its output polarity.

If the charging circuit is inside the battery-operated-device, there is a small chance that the external charger circuit could damage the device, but my guess is that it wouldn't be damaged.