Would すこし not be better to use than ちょっと in this situation (bc the amount is referring to "being busy")? Thinking of すこし as a type of quantitative measurement, the person who is busy does not have as much time
Yes, technically, watashi is the subject, however the focus in this case is 'tomorrow' Try thinking of it this way "Tomorrow I am busy" rather than "I am busy tomorrow" and that might help going forward. :)
ちょっと seems to be more common in spoken speech, and to my knowledge is used when speaking to both friends and strangers. It is also the better choice as the system doesn't specify to whom we are speaking.
If you actually want to include 私 instead of omitting it, it would sound better to say 明日、私はちょっと忙しいです。I still think the original sounds better though. It sounds more like you're saying "tomorrow's schedule is busy."