You may be having trouble finding things because calculus done with full rigor is called "analysis". The classic introductory-but-extremely-dense text is Rudin's _Principles of Mathematical Analysis_. If you're reading on your own, though, I expect you'll get a lot more out of a friendlier book; my first analysis course was taught out of Strichartz's _The Way of Analysis_.

N.B.: Most everything in either book will be completely new to you. 1st-year calculus courses have absolutely no interest in teaching you what goes on under the hood; there's just no time. But I think the analogy to Feynman's lectures holds up pretty well.