Yes, linguistic intuitions are an example of thoughts arising by a process inaccessible to consciousness.

The problem I have with the concept of "intuition" is that it's a non-apple sort of thing. It means, "I don't know how I know this", but has no implications for what the mechanisms really are. So I don't see a natural division between "intuitive" and "logical" thinking. Stuff that you're aware of and stuff that you aren't are both going on all the time. The boundary between them can itself change. Even for the stuff that you are aware of thinking, you aren't aware of the mechanism underlying that. However wide the circle illuminated by awareness, it always has a boundary, beyond which is non-awareness. Is there a good reason to suppose that essentially different mechanisms are in play inside and outside that circle?