If I sing without the melody, won't I stand a higher chance of missing the notes and not being in sync with the others?
Depends on your personal listening/singing style. B)
I actually Always sing these songs without a melody playing. I tried it a couple of times with the melody but I find it very distracting. I get much better results singing with no melody playing at all. Just emmerse yourself in the music and learn the song well enough that you can get the timing the way that feels right to you. If you have trouble with pitch, see if you can learn when you sound either "in tune" or "out of tune" with the music, and work at it until you find the note you want. Sing along to the original a little bit before hand, so you know what note the song starts on.
If you're off a little bit on entrances or timing, I can fix that in the mixing stage, for the most part. I will be spending a lot of time and effort on mixing for this song, and I have more sophisticated software than I had been using in the past.
Bottom line, do what YOU find easier. I prefer singing with no background melody, but then I also make sure to know and understand the song before I start singing.