Along the lines of Niko1978's answer, but less aggressive.
It appears you're trying to work yourself up through life and were handed what looked like a solid opportunity of advancement. It is entirely understandable that you trusted a verbal agreement with your current employer that didn't quite take shape. It really is 'shame on them!'
Moving forward, commit this to memory. 'Fool me twice, shame on me!' Promise yourself that you won't commit to an offer until it's in writing. And always make a copy for yourself because you'll never know when you'll need to refer back to it.
Secondly, you got burned. Approach your manger in a calm manner and remain nothing but professional. It sucks, but in hindsight going off, storming out, and ultimately quitting only burns bridges. That forces the time and effort in recreating those potentially solid references somewhere else. Present what you were offered and calmly ask if they are willing to fix their end of the agreement.
Finally, regardless of outcome bar some honest HR mistake, find new employment. I've been in your exact same shoes. If you're not in a position to put in your 2 weeks this exact second, do it passively until you find a new job. You should not need to put up with this. A dishonest company or a dishonest actor within a company will eventually make your life hell. If this is the case, then it's time to move on.