> Having studied Russian in college, I assumed that all Cyrillic script included a line over the т, because otherwise readability goes to hell.

It's entirely optional. In practice, there are very few words where т/ш pairs are needed for disambiguation.

> Did you end up mixing script in your native handwriting inadvertently?

Not really, but that's mostly because I have a somewhat personal style of English cursive, cobbled together from German and English hands.