I don't think there is any universal "intuition" to tap into, aside from that which comes from practice. You perhaps need to explore different physics texts in the electromagnetic department. I for example loathed Jackson as a learning text: it is comprehensive and useful as a reference for refreshing knowledge, but not good at conveying it. Volume 2 of the Feynman Lectures is a text I found excellent for a first course in electromagnetism.
There are two approaches I would take.
Geometrical Definitions of Nabla Operations
You mention that you are comfortable with Gauss's divergence theorem and Stokes's theorem. I would go back to the visual demonstrations of them as given in a good textbook - I recall "Vector Analysis" by Murray R. Spiegel having some wonderful diagrams along these lines - thinking of them as statements flux through surfaces and the relationship between this flux and the divergence within a closed bounding surface / line integral of the line bounding a surface and what happens when you shrink the surface / volume to infinitessimal size. This does two things: it gets you near to the original, integral forms of the equations of electromagnetism but more importantly it gives you a co-ordinate free way to define the divergence and curl of a vector field: define the component of the curl in a given direction as the limit of the line integral of a vector field around a loop in a plane normal to that direction divided by the loop's area: Stokes's theorem then becomes merely a proof that this procedure is well defined: the rest, i.e. the quantitative part about the flux of the curl through a loop being equal to the line integral around the loop then follows by definition. Likewise for the divergence (think of it as the nett flux out of a volume divided by the volume). Once you grasp the divergence and curl as definitions of geometric objects and not as collections of components, this might help. As an aside, you can see that the curl is defined in general in terms of a planar element of a surface, not the unit normal. The unit normal works in three dimensions - there is at most one plane through a point with a given unit normal in three dimensions - but you must think of the plane in higher dimensions as there is more than one plane through a point with a given unit normal so the unit normal is no longer useful in defining a plane.
Differential Forms
I think your approach of learning about forms is a good one and, notwithstanding frustrations, one you must persevere with. New paradigms like this are often good to pick up, mull over, then leave aside for a while before coming back to after you have processed them for a few days. Once you have thought about the nabla operator as I describe above, forms are a fairly natural step. Here you learn that the Stokes and Gauss theorems are two aspects of the same thing: the fundamental theorem of exterior calculus, which is simply a generalized form of Newton's fundamental calculus theorem. Two good texts here are:
Bernard Schutz, "Geometrical Methods of Mathematical Physics" Chapter 4 "Differential Forms"
and
Roger Penrose, "The Road to Reality" Chapter 12 "Manifolds of in n-Dimensions" and Chapter 14 "Calculus on Manifolds"
Lovelock and Rund, "Tensors, Differential Forms and Variational Principles"
This might sound crazy, but once you have battled with electromagnetism for a while, something like Chapter 4 "Electromagnetism and Differential Forms" of Misner Thorne and Wheeler, "Gravitation" or Penrose's discussion in Chapter 19 "The Classical Fields on Maxwell and Einstein" of "Road to Reality" are wondeful summaries to mull over. But you must learn a lower level approach first, because both these discussions move fast and do not prove much of what they say, but what they do say ties everything together beautifully.
