I think a lot of your problem might be solved by just a little bit of extra math, plus thinking really deeply about the material, rather than adding a lot more math. I was in a similar situation to you, but I took it to an extreme and got a PhD in math, but it ended up being a bit of a wild goose chase for me, since I am really more like an engineer or physicist at heart who is more concerned with nature/technology/applications than mathematical abstraction. Some of the math I learned was helpful, and I did get to the point where I understood a lot of things better about engineering math, but a lot of that had more to do with thinking about it more deeply on my own initiative than with studying all these big math theories, which I was required to do in my coursework. One big thing that did happen is that I became really good at linear algebra, and if I ever get back into engineering, I think that might be the number one thing that came out of the math PhD. Another benefit was the challenge, which makes a lot of things just seem easy by comparison.
For example, I think that complex analysis should precede differential equations - my curriculum lacked the course all together.
A whole course in complex analysis seems like overkill to me for basic ordinary diff eq. It's true that a lot of people will come out of high school with too poor of an understanding of complex arithmetic for diff eq, but that's chapter 1 of complex analysis, not the whole course. If it's PDE, then you might want to know complex analysis for some things, but I think it's not crucial for a first course.
Looking back at my experience as an EE major for a while, I found that a lot of the weaknesses in my understanding were corrected by more advanced courses and just reading chapter 1 of Visual Complex Analysis. For example, I took some circuits classes and initially, I didn't really understand phasor analysis, beyond being able to plug and chug with it because I didn't understand complex numbers well enough. Reading just that 1st chapter of VCA seemed to fix that, with a little extra thought on my part, applying it to circuits. That would also help with diff eq.
A lot of people probably don't understand the number e and its role in diff eq, despite being able to plug and chug with it because they take it on faith that the derivative of e^x is itself. I took care of that by figuring out on my own how to construct a function whose slope equals its height the first time I took calculus. It took me quite a bit longer to realize why it was actually the number e raised to the power x, rather than just a function, which you could call exp(x), and e = exp(1), by definition.
Another thing that bothered me was the way Fourier series were sort of pulled out of a hat. Historically, the motivation for that comes from the wave equation and vibrating strings. Any nice enough function pushed along the string will satisfy the wave equation. If you look at it from another point of view, though, suggested by studying oscillations, you arrive at the viewpoint that the motion of the string is a superposition of sine waves at frequencies that are multiples of the lowest, fundamental frequency. Put the two viewpoints together and voila, it seems as though any nice enough function can be represented as a sum of sine waves. This is the sort of thing you might read about in a book about the history of math or a physics book about waves and oscillations or classical mechanics.
Those are a few examples of conceptual dissatisfactions I had as an engineering student. Probably, I've covered a significant portion of it, actually.
