Mechanics and Fixing Any Questions Answered

There's no 'right' way to move the lever; it should just work regardless - with STIs it's possible to brake and shift in one movement.

Sounds like the grease inside is starting to go gummy; it happens after a decade or so. I'm not familiar with the internals of the later type, but the fundamental mechanism can't be too different from the old style, in which a small pawl pivots on a little pivot about 1.5mm thick, actuated by a tiny hairspring.

There's one of these for each shift direction, and when the friction in that pivot becomes too high, the lever doesn't reliably engage with the indexing mechanism because that pawl moves too slowly, until it worsens to the point where the hairspring can't overcome the friction at all, and the lever swings without ever catching the mechanism.

On the newer type of STI, where both cables run under the tape, it's easier to expose the mechanism; there's a convenient little hatch. I'm not too familiar with these, so I'm not sure how visible the engagement pawls are once the cover is off, but if you watch the mechanism as you shift it up and down, it should be possible to find the part in question; it'll be transmitting the lever movement inwards to the guts when it works, and not when it doesn't.

Best to do this with the lever off the bike, but at a minimum I'd disconnect all the cables and take the bars off at the steerer clamp. Remove the hood from the lever too, of course - the rubber doesn't take kindly to solvents like WD40 or whatever.

You need to point a squirt of solvent right at the sticky pivot and work it back and forth until the gunge is dissolved, and then chase it with some oil (use something a bit thinner if the weather gets very cold in your region).