Generally in advanced calculus I avoided the discussion of, and much connection with, geometry, Stokes theorem, and exterior algebra. So, I avoided Buck, Fleming, Spivak, and of course also, now in English, Henri Cartan, 'Differential Forms'. I even avoided the classic applied advanced calculus text, long used at MIT, Francis B. Hildebrand, 'Advanced Calculus for Applications'.

For Loomis and Sternberg, I agree with you, and have both the hard copy and the PDF.

Since I've mentioned such advanced calculus, I will try to save many students: Students, there's a secret. The secret is that vector analysis, Stokes theorem, etc. are important in physics and engineering; they will also be important in computing when computing concentrates on such physics and engineering. But still mostly what you will find in physics and engineering is vector analysis much as it was done in the 19th century which the late 20th century math departments liked about as much as a skunk at a garden party.

If you read the modern treatments, complete with differential forms, then you will be at the head of the class in an advanced class in general relativity (e.g., Misner, Thorne, and Wheeler) but will still be lost in much of old physics and engineering!

So, what to do? Sure, go to Tom M. Apostol, 'Mathematical Analysis: A Modern Approach to Advanced Calculus', Addison-Wesley, Reading, Massachusetts, 1957. The good thing about this book is the lie in the title -- it's mostly a 19th century treatment and not "modern"! So do whatever you have to do to get a copy. And get the 1957 edition and NOT a more recent edition where he omitted the 'good stuff'!

Then, in about 20 pages of the sweetest dessert you ever tasted, with line integrals, conservative force fields, and potentials, volume and surface integrals, nice stuff like that, you will find a charmingly clear presentation of what you need. Right: The treatment is not up to the precision of Rudin and actually needs pictures. Still it's what you need for much of physics and engineering. It's, uh, 'intuitive' math; trying to make that material as precise as Rudin could take you, well, a long time.

And it's EASY -- can take it with a couple of beers and have a really fun evening. Then don't tell anyone where you learned it! Besides, at its core, it's just nice uses of the fundamental theorem of calculus you saw in freshman calculus! Did I mention, it's easy?

The key point about Rudin's 'Principles' is the care with which he covers the real numbers, compactness, continuity and uniform continuity, sequences and series, and the Riemann integral (yes, patched up with the Stieltjes extension which isn't much different). So, he concentrates hard on the foundations. For someone like the OP, getting those foundations solid is likely more important than rushing into many of the more famous topics in 'advanced calculus' -- Fourier series, the heat equation, Lagrange multipliers, vibrating strings (boundary value problems), the Navier-Stokes equations, series solutions to ordinary differential equations, etc.

While I like Rudin, 'cut many of my math teeth' on Rudin, and really like some of his treatments of some topics, I omitted some notes on how to read Rudin; some such notes could be helpful. In particular, Rudin has some places where it's easy to get stuck, and students should be advised not to get stuck (don't assume that just because you can't see how to solve some one exercise must be missing something important) and if necessary just to look for other sources, ask for help, skip over and come back, or just f'get about it. Rudin was one of the best writers of his material, but he was not perfect, varied, got easier to read as he wrote more, but still is relatively severe. Due to the severity, there have been some people, e.g., at Courant, who didn't like Rudin!

As much as I like the real half of his 'R&CA', he gets a bit severe and obscure in a few places (his novel and surprising but long and 'unstructured' construction of Lebesgue measure and his work on regular Borel measures); net, for most students it would be good to read Royden first or in parallel.

Rudin has two exercises that can slow people down: (1) Every closed set is the union of a perfect set and a set that is at most countable and (2) there are no countably infinite sigma algebras. Both exercises require paying attention to what is countable versus uncountable. The first one I worked on about 14 hours a day for two weeks before someone mentioned 'uncountable' at which time I got it in about 90 seconds. The second one took me a long evening, but I was the only one in the class who got it. For the first one, eventually Rudin included the hint. Students: Don't get stuck on such exercises.

For Rudin's 'Functional Analysis', I nearly went to Brown's Division of Applied Math but at the last moment went to Hopkins instead. Brown was using Rudin's FA, so I got a copy and at Hopkins asked for a reading course in it. Alas, the prof had never heard of that book and declined to participate! So, Rudin's FA, along with his 'Fourier Analysis on Groups' or some such are still sitting new on my shelf as I write software!