Originally Posted by Trakhak

If you're determined to spend the money on changing out the shifters and whatever else would need to be replaced, then, never mind.

But:

(I) Shimano brake/shifter components are notorious, not for wearing out, but for shifting more and more poorly as the internal lubrication gradually thickens and turns to sludge. The fact that you had two exhibit the same symptoms all but guarantees that the problem is not that the parts are worn but that the shifters need to be purged of the remnants of the original lubricant and relubed. (Search YouTube for guidance on how to perform the purge.)

(2) If you decide that the elaborate purging process seems like too much trouble (that's what I decided about my gummy Ultegra 10-speed brifters) and you're open to the idea of installing non-prestigious but worthwhile components on the bike, you might consider a pair of, e.g., Microshift brifters.

I paid about $80 for mine, and I greatly prefer them to the original Ultegra ones. They snap into each gear much more positively and satisfyingly than the Ultegras ever did.

The combination of a thumb lever and a blade for the fingers, while very different from the two-blade Shimano system, is very easy to get used to. And the levers are clearly going to last far longer than the Shimanos, with minimal or no maintenance.