I find that grouping things together in a way that you can understand them helps you remember it more easily. For example, I had to memorize a list of 53 pKa's for Organic Chemistry II (40 on a sheet of paper and 13 from a table), so I found relationships between each acid and grouped them together.
Maybe you could analyze the song you want to try singing, break it down to what words each section involves, and just think about how many words into each verse you have to know before you can figure out the rest without explicitly remembering them. Find the differences in context between each verse, and find the choruses. Form a general structure to the song in your head.
There have been times when I've messed up badly, and honestly, no one really noticed. The first time was my audition for playing the piano at the talent show in my senior year of high school. I knew it was pretty bad, but I just kept going (and made one more mistake), and then finished, and I still got on the talent show. No one even asked about anything remotely related to the mistakes. Now let's go back to sophomore year. I played Mario for the talent show, and I knew I made a mistake live. I randomly forgot what I was playing right in the middle of the performance, and I paused for about a second. I went with it anyways, picked a note to fill in the gap, and continued playing where I left off. I think my mind went blank for a moment, but I don't know if I just got lucky or I improvised my way through (aka BS). I feel like that would help. Of course, it may just have been easier because I was shorter than the piano when I sat down, and the piano was facing perpendicular to the audience, not to mention the stage lights made it impossible to see the audience.