Just some observations and ideas:
It won't matter what you sing though if you lack confidence and technique. You'll sound like shit through a U87 just as well as an SM57 if you don't sing a part well.
Try out different styles of singing, pretending you are a different singer so you don't feel a lack of confidence.
Put some of that money towards some singing lessons from a coach.
Read a bit on how to fix certain problems (ie not singing the first letter of a word if it's a consonant when punching in on background vocals so you don't come in the wrong time as the lead. Or just edit that out to leave the lead clear. Work on problem Eeees so they don't sound harsh. Etc. I recommend doing a take, listening to it, and evaluating the areas you cringe at, and mentally think how you could have sung it differently to make it not cringe-worthy. Like a change in word? Pitch?)
The SM7B is for a powerful singer. Not soft vocals. You want a condenser mic. That Warm Audio sounds like a fantastic deal. But you gotta know if that mic is good for your voice. Sometimes it's a mic-per-type-of-song deal. I sang one song with an SM57 that sounded better than I expected. All other mics didn't do. Never found another song the SM57 worked with since. And your recording chain has a lot to do with it.
I second the Neve 1073 plugin idea. I use Universal Audio and it's a great additional to warming up a vocal take and smoothing out harshness. Learn to do some slight EQ drops on your own vocals to suit.
It's a LOT to making a vocal you like. But It starts with coaching. Runs through many gear trials and signal pathway trials with different gear/plugins. and then ends with smart editing. IF you are doing it all yourself.
You COULD invest in some studio time and tell them your interest in a cheaper mic, and have them use various ones on you and tell you what they used, what EQ curve they used on you, etc. Cost you a few, but would be well worth it.