You seem to know the basics, if anything you should just watch Mengsk replays or keep playing to get used to it, I think you should’ve just gone for rocket troopers than sky fury, sky furies are more like giant killers than a dedicated anti-air unit even though they do the job well enough.
Here’s the replay, try to look for the first 7 minutes into the game especially, rest is optional.
Remember that this map is the one that you can get away being super greedy, other maps the second bunker goes to the ramp to defend attack wave.
So here’s how it goes:
This match I tried to mimic a low lv Mengsk to make it a bit easier to understand, so I removed my mastery points and prestige, didn’t used nukes nor Dogs of War to just show how it goes for the most part (you have no idea how my fingers were itchy to pop Zerg at 100 mandate lol), and tried to tone down the APM, though that didn’t work very well.
The reason why witness is very important for Mengsk is to spam calldowns and you will need it especially for Chain of Ascension where lack of detection can potentially kill you, you will learn to hate lurkers exclusivelly.
At lv 1-4, Mengsk is quite weak because there are no ways of increasing mandate generation, but at lv 5 and beyond you can do the build order like this replay, of course on contested expos you just focus on witness and medivacs, RoK you make all of your buildings at ramp to prevent surround on bunker.
In this replay I should’ve gone for ghosts instead of tanks, the emp also stuns mech units which is so good, tanks do well vs protoss but emp is easier to use for me.
Lastly, keep trying to use hotkeys so you avoid to contantly look at base, I tried to play low APM but if anything I made it low EPM Facepalm