No, I never call a gadol hador with a shaila. They have far better things to do than listen to me.
If I have a shaila, I ask my rav. Were he to advise me to escalate it to someone else I would then do so. But no matter how pressing or how complex, the question goes to my local Rav first. I trust him to say “this is beyond me, please go ask X” if it is beyond his knowledge or areas of expertise. I have no reason to ever call a gadol and ask him my question first. (And, IMHO, nor does anyone else, unless the gadol happens to be their personal rav.)
The Wolf