I don't think anything would be messed because you jump between the two DLC's. I have done that on my current playthrough. I have only discovered Miraak's temple and haven't touched anything else on Solstheim but I am halfway through Dawnguard. I haven't found anything that suggests the game messed coz of my jumping DLC's.Hope my experience helped.
I am not that sure about the DLC's conflicting or making a difference on which you complete first. I hope they don't. of course if you complete DB first you get to ride a Dragon relatively earlier.
I am not to be trusted regarding the matter of the game messing. I AM new here as well as to the concept of solving out Skyrim problems.