I was able to find some sufficiently small hose clamps that fixed my air leakage problems, but now I'm having trouble with the accuracy of the pressure sensor. Here is some of the data I got during my last test session:

SG test solution -> ADC counts (0-1023)
1.000 -> 220
1.100 -> 240
1.062 -> 237
1.028 -> 233
1.000 -> 227
1.100 -> 244
1.062 -> 237
1.100 -> 242
1.000 -> 223
1.000 -> 226

So as you can see the trend is generally correct but there is too much hysteresis in the values. After perusing the datasheet for the sensor it seems that the overall accuracy is +/- 5% worst-case, so the sensor is performing much better than its spec. But it's still not very useful for my purposes.

I've been working with the pressure sensor again. I finally got a very ugly prototype of a hand-held meter built. The basic idea is that I have the pressure sensor wired to one of the analog inputs of an arduino, and then the arduino sends sample data out via serial to my PC.

I got some pretty good readings that look like I'll have a resolution of around 0.005 SG (which is usable, I think). The only major problem I ran into was that air was slowly leaking out the tubes that were in the water, which means the seal I have between the tubes and the pressure sensor ports isn't good enough. So the next thing on my list is to come up with a way to seal the tubes better.

Here are some pictures of the prototype: