I am not a singer, but a lot of this applies to me, also. There is something different between "memorizing" a song/tune, and "having" it. When I was in secondary school playing in Violin Competitions, I had to Memorize pages long concertos,including memorizing the rests. Today,over 30 years later, I can play maybe 6 bars of 2 different concertos by memory. A whole concerto? Forget it. "Knowing", or "having" is different. If I live to be 200, I will not forget how to play Redwing. It's just there in the hard wiring.
Memorizing is WORK, doing one phrase at a time. Having it means you have heard it so much you can do it automatically. I prefer to Have a tune so well that I can interact with my audience. Wink, dance, whatever.
I do sometimes play with my eyes closed just to filter out some of the distraction so I can listen to the mix better, but when on stage, I prefer not to. It is disconcerting to an audience to watch a perfomer who seems oblivious of the audience that (usually) paid, and looks like s/he is asleep at the wheel. (I also tend to stick my tongue out when concentrating........other fiddler has been told to kick me when I do that on stage!)