The problem was trying to carry the yarn all the way around. One way to do it is to knit if flat and do the name intarsia.
If you're really averse to knitting it flat, here's how to do it in the round--I'll try to be as clear as possible.
Knit with the white until the first of the lettering, use intarsia or stranding for the letters depending how far apart they are, and work across with both colors for the first row. Drop the green and work around in the white. Now, on the next row, you're green yarn is in the wrong place, right?
Work to the end of your name knitting the white stitches, slipping the green stitches.
Turn your work, and purl the green stitches as you slip the white. Turn back to the knit side.
Slip everything back on the right needle and knit around with the white until you get to the name again. This time your green will be on the correct side, so you can knit across.
Repeat these rows until the name is finished.