I find one of the biggest obstacles is kidding yourself you know it when it's almost there but not quite and this is true especially when it's a "song we all know".
I think it's a good idea to lock onto the first verse alone and hammer it down. Beware moving on too fast. Because it's the first verse it feels familiar, like an old acquaintance we take for granted but have never really got to know. I read somewhere to take the first note of the first line, sing it a few times then sing the whole line to this note. Do it a few times and then sing it with all the notes. Apart from really enjoying the release from the monotone, you really get a feel for the nuances of the line and it seems to stick even more than singing from the sheetmusic where it's already worked out for you so a vital part of the embedding process is missed.
Once the first verse is thoroughly learned the other verses will come much easier and if there are variations(tune wise) in other verses these will be really conspicuous against the backdrop of what you've put in place and so be more accessible.
I think this works. I know it helps me.