I've heard that more powerful amps would damage my factory speakers ('98 Honda Civic). Is this true?

Any amplifier can blow any speaker if you don't set it up and use it correctly.

The truth is that a more powerful amplifier gives more clean headroom than a less powerful one. You just adjust its gains down properly so that it does not overpower the speakers. A more powerful amp, when properly adjusted, is less likely to blow speakers than a less powerful one.

The adjustment, of course, is critical. Details are here. Scan the page for the words "Properly adjust the input gain level on the amplifier(s)".