"...it's my pet preference to sing it like that, too, when I'm singing to myself (say, while shaving in the morning).)
I think it is necessary to know whether this shaving is of the traditional 'cut throat', safety razor or donkey engine variety.
Also which part of the anatomy is being shaved, as this would necessarily alter the metre. A quick scrape is indicated when using Johnny Come Down To Hilo, but a slow deliberate 'we all shave under the CHIN' might be more appropriate to avoid nicks.
The term Hee-lo (yodelled) is called for when applying after shave :-)