Here's what I've done a few times.

1. If the tune's fiddly or weird, or there's variation between verses, or anything else 'interesting' is going on, listen properly to a recording of the song - play it several times in a row. You're not trying to learn the song at this stage; you don't even need to be concentrating on it. The idea is to ingrain the tune in your head, and the way the tune fits together with the words.
2 (essential). Sing it - while walking, washing up, waiting for a bus, whatever. If the tune's not there or it doesn't seem to fit, try step 1 again. The words probably won't be all there; if they are (maybe it's only a short song), go to 4.
3. Print a copy of the lyrics and put it in your pocket. Then sing the song some more - walking down the street, hanging out the washing, whatever - but stop and check whenever you go blank. It's not as laborious as it sounds - after a while words can lodge surprisingly quickly. (I learned "Hard Rain" this way.)
4. Sing it some more, but concentrate on what you're singing. Do you mean what you're singing? Have you got a feel for how the song fits together? Are you stressing the right words, i.e. the words you'd naturally stress if you were speaking the lyrics in conversation? (If the answer's No, can you stress the right words without screwing up the tune? And if the answer to that is No, can you keep the conversational stresses and make the tune fit round them?)

As for how many, unless you're working up two very different songs (Searching for Lambs and Glorious Ale, or something) I think it has to be one at a time.