First of all, if you can sing and are having trouble with this, either lower the difficulty or try listening to the songs online beforehand. This really helped me with Tangled Up in Blue especially.
However, if you're reading this solution, you probably aren't the best singer in the world. In this case, I find it's the easiest to actually not try to sing the song at all. Trying usually just ends in failure. Instead, know the basic tune of the song and just kinda groan out the notes. Similar to humming but keep your mouth open and make sure you're loud enough for the mike to pick you up. If you stay quiet enough, it's easy to hear the notes of the song and imitate them.
Finally, I'm not entirely sure if this would work or not but if you could find a vocal track of any of the songs without the instruments, you may be able to play that on your computer and put the mike up to the speaker. Not sure if that would work or not but as long as there aren't any instrumentals, I can't see why not. Or you could always check out the above solutions and just have someone else do it while you play another instrument.