I have both but to be honest Richard I've never used the CP!
There's a short video demo of a CPL here on YouTube....
http://www.youtube.com/watch?feature=fvwp&NR=1&v=MNzXrTKEMWg
It is very obvious to see how easily the blue parts of the sky can be tweaked with the polariser while leaving other parts of the scene relatively unaffected. You don't even need to fit the filter to see the effect - just twiddle it in front of the lens or even your own eyes without a camera.
That said, I'm not sure how that is going to help the OP with blown skies, because you'd have to go some to blow out a truly "blue" sky.....
A white/hazy/cloudy sky like those I more typically find round my way is another matter, but one with which I don't think a polariser will help....
Returning to the original post, there have been several good suggestions, but without seeing any sample shots and especially back to back comparisons with the DSLR against the compact it is hard to know which solution(s) would be best, or why the DSLR did deliver such different results. Probably the most important thing to appreciate is how to read the lighting itself. As an alternative to overcoming the problem maybe it is possible to avoid it altogether, by changing the composition a little, shooting tighter or from a higher perspective, or turning around to shoot from a different angle completely.
Just as a small example of what I mean, here are two shots of a castle, taken a couple of minutes apart, but from completely different directions. One angle of attack presents a scene which is very easy to shoot, with an undemanding dynamic range. Moving around the castle you have the castle in its own shadow, but the clouds/sky now backlit by the sun, making the castle very dark and the sky very bright. If you expose for the castle the sky will be blown. If you expose for the sky then the castle will be too dark. Filters might work, or blending shots, but perhaps it would be easier just to pick a more forgiving shooting angle in the first place.
The change in aperture value and shutter speed are not individually important here, but if we weigh up the differences overall in exposure values the darker shot has an exposure 1.7 stops dimmer than the brighter shot. Not only that, but we are on the darker side of the castle, so that's kind of counter intuitive, and the problem is all because of the sky. Backlit sky is always going to be tricky. If you don't have the tools to deal with it (filters, fill flash, HDR) then my advice is to try to avoid it.
When I have no choice but to shoot against the light, with a bright sky behind, then I will usually meter for the sky and bring up the foreground in post. Here's one such example....
The red areas show tiny parts of the sky are blown, but not enough to ruin the sky and the details within it. By shooting at low ISO it is not a problem to increase the brightness of the foreground and still retain low noise in the shadows.
There are times when backlighting can be very appealing for a subject, but not if the backlighting includes an overexposed and blown "white" sky. Prevention is sometimes better than cure, so see what you can do to avoid it completely rather than trying to hammer it into submission. So, for example, if you're on a family holiday and want some shots of the family members you probably have some options about where to position them, and if you're shooting more candidly, maybe you have some options about where to position yourself.
p.s. just as a simple exercise, on a sunny day go outside and look around you, turning through 360 degrees. You'll probably find that facing away from the sun the sky does indeed look a lovely clear blue, even if there are a few clouds dotted around. But as you turn ever closer towards the direction of the sun you may see the sky starting to whiten, until as you are lined up with the sun there really isn't much evidence of a "blue" sky at all. At least that's the case where I live, just on the eastern outskirts of London with "dirty" atmosphere. If you're in the sticks or up in the mountains it will probably be a different story.
EDIT : I just popped out to grab a couple of shots to illustrate the point above. The camera was set to Av mode and without EC it delivered exposures only 1/3 stop different when pointing away from and towards the sun. There was no filter fitted.
Pointing away from the sun the blues of the sky are very deep and rich and the foreground is nicely exposed. Pointing towards the sun the areas of sky between the clouds appear hazy and whitened, reducing the saturation of the sky and brightening the "blue" areas. The foreground subjects are in their own shadows and thus rather underexposed individually, although as a scene overall the exposure probably looks reasonable. However, if you wanted to bring up the exposure for the foreground within the camera rather than in software there is no option but to watch the sky become increasingly blown out.