I prefer
Corinne over both of those but since it's not an option then I'll say
Corrine. But in all honesty
Korrine doesn't bother me. Sometimes the K replacement spellings irritate me and others they don't, in this case I don't really mind.
I've always pronounced them as ko-RIN.
"Substitution mass confusion
Clouds inside your head
Involving all my energies
Until you visited
With your eyes of porcelain and of blue
They shock me into sense
You think you're so illustrious
You call yourself intense"
-Bye Bye Love, The Cars