Cayene - when you say, "blown out", do you mean that the whites are clipped? Specifically, have you lost detail in the highlights? In Resolve's scopes, this would show up as a straight horizontal line at the top of the Waveform monitor or straight vertical lines at the right of the histogram.
How did you expose in camera? Were you relying on the 5D's exposure meter to hit the center (or 1 stop below, when underexposing)? I have found that Cinestyle really needs to be exposed by ignoring the meter, looking at the histogram and overexposing (commonly called "expose to the right" or "ETTR" for short), but being careful so that nothing important is clipped (i.e., no vertical line showing up on the right side of the histogram). This is really easy when there is no highlight in the scene that SHOULD be blown out. If there is something in the scene that should be blown out, such as shooting directly at a light source or a reflection of a light source, then that should show up as clipped. Once I got used to using the histogram, I could figure out what was clipping and what wasn't. Basically, the histogram gives you more info than the light meter and you expose what you want.
When using ETTR, the image will look like crap prior to color correction. But, after you bring down the shadows (and perhaps the mids) and add contrast, it should look very good, assuming no important highlights were actually blown out or clipped. In essence, you are pushing the noise to black by doing this.
By underexposing and then lifting the exposure in post, you are adding digital gain, which will increase the noise, not decrease it. You want to go the other way: overexpose and then decrease the exposure in post. This reduces noise. As I said above, the catch here is to not overexpose so much that you lose the highlight detail.