Howdy there! I'm a fellow gold terran pushing into platinum, I think I can help with this. Hopefully. I think that just over time, with practice, your play will slowly but surely become faster. This of course depends on how much you want to practice, but if you practice regularly (2-5 games a day is what I've been doing) these things will become automatic and your multitasking will become better. When you've been in a situation enough times to know exactly what to do, you'll do it and have more "thinking time" to figure out what you want to accomplish on top of, say, dealing with an early gateway push, as well as more time to micro and not miss depots or production. Optimizing your builds can help this exponentially, so you automatically expand, build extra racks, etc., without having to think about it or guess in regards to the timing.
Also, Day said something in a fairly recent video about low level players figuring out how to win in general as opposed to figuring how to win that specific game. Maybe incorporate this process into your play? Lately, I haven't cared if I lost if my macro was good and all I need to do is work on micro, but I will care if I come across a new strategy (the sentry-warp prism play where the Protoss warps in sentries in your base and forcefields was unknown to me until about three weeks ago) and lose. Then, I'll in-depth watch the replay, try to figure out what I did wrong, as well as figure out how the build works and what it would be vulnerable to. Learning what enemy's builds are bad against is just as important as knowing what your own weaknesses are.
Hopefully that makes some sort of rambling sense?