It sounds like you want to knit the letters intarsia style, which is usually not done in the round for the specific reason you stated.
Duplicate stitch after the fact will work better.
There is a way to work them in the round when both colors are carried all the way through, and since it's a Christmas stocking and nobody's toes are going to get caught, you can try it, though I'm afraid the red might show through the white.
Knit across the first row of the letters with both colors, stranding the yarns loosely, but leaving the red yarn where it is. Make sure to bring the white up from under the red to prevent a hole.
On the next round, your red yarn will be in the wrong place, but the white will have been knit around to the beginning of the letters.
Knit across the white stitches, but slip the red ones. Then turn your work and purl the red stitches and slip the white ones. Then slip all the stitches back. Your white will be in the right place to continue, and your red will be in place for the next row.