My double knitting pattern is confusing me
If your pattern tells you to unravel your caston, it should have also told you to start with a provisional caston. A provisional caston is a method that allows you do easily unravel the caston, turning the beginning of the work into live stitches. Here's a link to one method of provisional caston.
It sounds like maybe you didn't do a provisional caston (oops). If that's the case (regardless of whether it's your mistake or the pattern designer's), you have three options to fix it.
Option one: Pick up and knit stitches from the caston edge, just like you would pick up and knit stitches along any other finished edge. Here's a source with a photo tutorial. This is by far the easier method, but it will create a ridge or seam that will be visible on the wrong side of the work. The ridge might be completely invisible from the right side of the work, or it might not. It might also be less stretchy than the rest of the work. I recommend you try this method first, and see how it looks and stretches. If you're not willing to tolerate the ridge, rip out your picked up stitches and use the next method.
Option two: Start by threading a knitting needle through the first row of stitches. It might help to use a smaller size of needle than the one you cast on with. Now find the end of the yarn and start undoing the caston edge. This means finding the next bit of yarn above the loose end, then pulling that bit of yarn until the loose end comes through whichever other bit of yarn it's looped through. Repeat. It will be a slow, painstaking process, and you will quickly learn why special provisional caston methods exist.
Note: it's probably not worth the effort to unpick the caston edge. You will very likely get several stitches in before you want to say &quot;F--k it!&quot; and give up. At that point, you can just leave the rest of the caston edge as is, and use the stitches that you picked up already. Some of your stitches here will have a ridge on the wrong side, and some of them will not.
Option three: Insert the knitting needle through the second row of stitches. Cut the yarn through a stitch in the first row, and unravel that first row. (I know you said you don't want to do this, but it may be your best option at this point. Try the other methods, and come back to this one if they others don't work for you.)

Yes, double knitting is worked with both sets of stitches on one needle. You can set it up by putting all the stitches on one needle, alternating one stitch from the front needle, then one stitch from the back needle, etc. Or you can start with the stitches on two needles, and work the first row, alternating one stitch from the front needle, then one stitch from the back needle. At the end of this row, all the stitches will be on one needle.