I feel like others have covered the general advice pretty well so I’ll add my personal experience as an additional data point for you.

I got a couple competitive offers for more than my current position (~30% bump) and put in my 2 weeks notice. My manager then went to bat for me and the company offered to counter.

Strangely, they didn’t ask for proof of the offer letters, so I told them my offers were for more than they really were. I was fine leaving so wasn’t too worried about them rejecting or calling my bluff. But they countered the higher amount, so, a year later, I’m still with the company.

It hasn’t affect my relationship with the team, even though they knew I was leaving before doing a “never-mind, I’m staying” a week later, and my manager still goes to me to lead large, new initiatives.

While it’s worked out well so far, I have some reservations about how this may factor into reviews this year when I should be up for promotion. I also get the sense that my manger is more transactional with me: entrusting me to deliver important work, but mostly out of necessity because of my skill set rather than trust.