Spam clicks. It sounds stupid but if you press it enough times it must release adrenaline (epinephrine) or something that makes you act faster. After a while of getting used to just clicking, start using those clicks to actually check different things - unit health, reface weapons, zoom around the map etc. After a while you get used to your clicks actually having a purpose.
Try to conscriously think of the placement of every single unit. During a battle, micro each unit to a separate piece of cover. You can select all your units and use tab to cycle through them one at a time if your clicking skills still aren't up to scratch. After practicing that for a bit, expand the area so you now control more than just one battle, you control your flanking units and support units as well. After that, expand to controlling all units on the map simultaneously.
Remember that the tactical map, minimap and the unit tab on the right of the screen are useful if you forget about units often. Make it a habit to check these as often as possible. The minimap should be checked every few seconds and the tac map/unit list should be checked every minute if you're not good at remembering where you put your units. Check back to your base every 30 seconds or so to make sure the units there are healing/reinforcing properly. Don't waste too much time here. Select the unit, spam R as many times as you can and then move out. Reinforcement of a full army shouldn't take longer than 10 seconds. When I reinforce I check back at base, select everything, tab through while spamming R (takes 10s max), look around the map again/give orders (10s), return to base and give the orders to move out. Watching them reinforce is wasted time.
Micro improves with practice. There's no magic thing you can do to suddenly be awesome at it but by tricking your brain into over-excitement (even in dull periods) you can train yourself to respond quicker when you do need it. You don't have to overthink unit placement when you put them in cover, just move to the first piece of cover available. The action is what is important, not the reasoning. The reasoning is something separate (known as "game sense") that also comes from practice.
If you get into an argument with me, you're wrong.