Ah, I see... I was thinking you were talking about amplifying the RF.
Yes, the LM386 works as an audio modulation source for the transmitter, but it's not ideal. In my tests the audio was always a bit dampened. It works, but it's not the best. It's really simple to construct, so that's the tradeoff.
As for the LM386 getting hot, I'm not sure if mine did or not... It's been a while since I tinkered with that circuit. What I presented in the schematic was the bare essential parts to get it working. You might have better luck building the full LM386 circuit presented in the datasheet. It has a few more parts that might help with stability. Also, now that I think about it, one reason it might be heating up is the voltage input is too high. You could try to step the voltage down to the chip, which might help. Or, the current might be too high, and in such a case you might need to add in a resistor to limit the current at the input.