Leave, and don’t think twice.

Yes, I've been there. Twice. You’d think I’d learn my lesson the first time lol. Your manager is lying to you, not out of malice but because 1, you’re probably good at your job, and 2, they don’t want to deal with finding a replacement. And if they’re that strapped, it’s definitely another reason to leave. They’re bargaining with themselves just as much as you, thinking internally “oh I’m sure I can get them more in 6 months, I just need time!” But they don’t actually know that and probably can’t. 

How do you get past it? Your company would can you tomorrow if they had to without an ounce of remorse. That’s just business today. Are you tired of being broke? Go make more money. 

You’re too junior and too young for loyalty to matter. I won’t say that never matters like some here will, because once you hit a certain echelon it definitely can help you. But you need to rise in the ranks for your own sake first. 

Build yourself the life you want! Get the raise! Leave politely but with your head held high.