For me the two are inter-related.
They're inter-related in that they are opposites. Asking for evidence means you do
not trust your gut feeling, regardless of how strong, but instead, require evidence. That is what skepticism advocates.
If you were "skeptical" of this method, that would mean that you doubt that asking for evidence is the best method for determining truth. (I put "skeptical" in quotes in this paragraph, because I am using it in the sense purely of doubt, not of asking for evidence. Otherwise, you get trapped in the logic loop of asking for evidence of the utility of asking for evidence.)
Now I understand that you are saying that you use a mixture of both evidence and gut-feeling when evaluating the truth of a proposition. But are you really? So called "intuition" is not as simple as it might seem. In truth, your intuition may be based on experience, sensory clues, even subconscious analysis of evidence. Intuition becomes much better the more you know about something.
For example, you could take a person who knew nothing about horses which horse (of a group of horses he was seeing for the very first time) would win a race, their picks would probably perform no better than random. If you asked an experienced horse-racing fan the same thing, showing him horses he had never seen before and knew nothing of their records, he would almost certainly perform much better, simply because he would pick up evidence
just by looking at them. He might not even realize he was doing this and call it "gut feeling", but in this case, "gut feeling" is still based on evidence.
Is that person a skeptic, just because he is relying on evidence to make his picks? Not necessarily. A skeptic is
aware that he is using evidence and indeed strives to identify and evaluate that evidence. All things being equal, the skeptic and the non-skeptic gamblers would perform about the same if they were evaluating the same evidence. The difference would be that the skeptic could consciously learn from this experience and consciously apply it to future situations.
One of the advantages of skepticism is that, on the whole, skeptics, by consciously engaging their brains, will learn faster than those who use their "gut" as a repository of knowledge.