My advice is different. Work through a standard undergrad text on DiffyQs first. Not one with all the fancy connections to other fields of math that you know. But one emphasizing manipulation and problem solving and applications. Then after that, go grab some fancy book with all the grad school emphasis on proofs and Sobolev spaces and the like.

Because 90%+ of those books assume exposure to diffyQs first (in the way that "real analysis" typically assumes "calculus" exposure first). And it's not just about how books are constructed and how people typically learn. It's actually more efficient and you will learn more and deeper by learning the content first in terms of problem solving manipulation and later in terms of all the fancy stuff. It's pedagogically advantageous. The human brain is not a computer, it learns from imitation and repetition. This is why you can't teach a young gymnast a double back when they start.

A good, cheap book for self study is Tenanbaum and Pollard. 800 pages, all problems have answers. Covers the whole playing field. And even has a lot of non rigorous proofs.

P.s. I'm curious if you have a gap in PDEs also. Can't see how you can work with PDEs if you don't have undergrad familiarity with ODEs as many problems are solved by converting a PDE to an ODE.