Do you have to speak words borrowed from English as if you're using katakana?

Well, this really isn’t my expertise at all, so I can answer as a fellow learner who is a kindred spirit in terms of trying to come to grips with Japanese intonation.

I think a combination of awareness and good role models are probably the best.

By awareness I mean that you should not be satisfied with learning what a word means, or how you write it, but also ask yourself if it is accented (and where, if so).

By good role models I mean that you won’t get anywhere without listening to how this mess sounds. Because one thing is learning the intonation of words in isolation, which is all well and good. But another entirely different thing is to learn how to those accent patterns interact with one another when you are building a sentence.

I try to listen to as much Japanese as I can. I listen to a lot of podcasts, and some of them are in Japanese. I put some up on the ultimate resource list. I find that listening to those and shadowing is a good practice.

That said, intonation is still by far the thing that I struggle with the most.