So if you want more definition in your sky, you have to let less light in. On an overcast day, you still get a lot of light, so that can fool you into thinking you need to let more in.
Bump the ISO down to 200, open the F-stop a bit more (maybe to around 8 or 9) and then adjust the shutter speed a bit until you find the happy medium.
Also, I'd suggest looking into HDR post-processing if you really want to make the sky pop without losing the detail of the landscape.
Below is an example of photos I took in California and then the HDR result.
These all used the same ISO (200), F-stop (4.5) and focal length (18mm). The difference was how long I left the shutter open.
1/60
1/100
1/250
1/800
1/600
HDR result:
With HDR, I get greener, brighter grass and more definition in the foggy background.