The idea of the power adapter crossed my mind, as well, but then I remembered you said it did the same thing even when running on batteries. So, yeah, that generally rules out the power adapter. However, I am a bit surprised that a different power adapter worked, even briefly. With that burning smell you mentioned coming from the speaker, that's really pointing back to a shorted speaker or amplifier problem, in my estimation. It seems like a short is causing the keyboard to pull too much current -- so much that it exceeds the power capabilities of the AC adapter or batteries -- so that the keyboard then shuts down. The fact that you said the replacement adapter is rated for a higher amperage may explain why it worked briefly.
It is possible that this potential excessive current draw has damaged both of those AC adapters, so I would check them before using them in any other device. It is also possible that the AC adapters have a protection circuit in them to shut down when they are overloaded, so it may also be the case that they are fine. They would have to be tested under load conditions, because a digital voltmeter draws very little current, so they could read the proper voltage when you just hook them up to the volt meter, but then not work properly when you actually try to power a device with them.