I have a height map I've generated that I want to use to create an island map.
However, when I import it into CAW, it comes out shallow, with the land only a bit higher than the water.
I increased and decreased the maximum map height, but the change is very little.
Increasing the ocean level causes the island itself to submerge.

Here are the two versions of the height map I've tried. Could someone help me?

The contrast of the heightmap gives the depth- so the lighter the grey, the higher. But your second map had high contrast. Do you have the bit depth correct on the heightmap?

I'm having a hard time with that, actually. I managed to get it to work in CAW as a 32-bit, even though I know that's wrong.
In fact, that might be why I'm having this problem.
I haven't been able to find a way to convert to 16-bit PNG without Photoshop. Do you know any way I can do that?

Yes, there are several free programs that will deal with this. I use GIMP; Image>Mode>Greyscale and Image>Precision>16 bit Integer. Make sure you only have a single layer. That will give you a 16 bit greyscale.

That did it. I suppose we can say this is just another bit depth issue.

Well done for fixing it!