I probably shouldn't get involved with tackling this question. I haven't done any DK for a long time, except I just tried it again and it is coming out okay. I've only done the heart potholder and a couple of hats in DK, but that was quite a feat if I do say so myself.
I love the scarf you mentioned but never tried that.
Are you only working with 2 colors at a time and are they always one color on one side and the alternate color on the other side, doing the same thing but reversed in the colors? That is the way I've done it. You always work with both colors in your hands. There are two totally different techniques for DK one involves slipping stitches and one uses knits and purls. I have not done the slip st technique.
When you were trying to make the checkerboard pattern are you doing lets say 4 sts of red and then 4 of blue and then 4 of red again and then 4 of blue, all this on one side and the opposite side reversed in color? (because that is what I did
) And are you having St st facing out on both sides? If this is what you are trying to do here are my observations.
On the first row lets say you start with red on the side facing you and blue on the back side. Begin by holding both yarns in back and knitting with the red, then bring both yarns to the front and purl with the blue. The first stitch you work each time will be knit and it will be the one that faces you, then you purl the next stitch which makes it turn out knit on the outside on the back. So always knit the stitch with the color you want facing you and purl with the stitch you want in back. You don't really need to twist anything to make this all happen.
Here is how you do a checkerboard of 4 red stitches, 4 blue, and repeat those. Wyb( with yarn back--always move both the yarns front and back together) k1R(knit one red), wyf (with yarn front) p1B (purl 1 blue), wyb k1R, wyf p1B, wyfb k1R, wyf p1B, wyb k1R, wyf p1B. Now you have 4 red on the front, the next stitch will be blue on front so (wyb K1B, wyf p1R,) repeat 3 more times. Now you have 4 blue in front. So you are ready to switch to red in front again. (wyb k1R, wyf k1B) repeat 3 more times. Then switch to blue again. (wyb k1B, wyf p1R) repeat 3 more times.
What to remember is that stitches are done in pairs always a knit and a purl beginning with the color you want facing you as the knit and the color that will be in back as the purl.
~Also sometimes I end up with purl 'bumps' when the fabric should appear to be stockinette on both sides.
This is probably because you a purl or a knit in the wrong place. I think of it as if I'm making two pieces at once, the one facing me is all knit and the one away from me is all purl. If you make a mistake it is not that hard to fix it, just keep the sides separate and fix each side as a unit.
I did a Google search for "how double knit" and some videos came up on youtube I think as well as some other things. I don't know if they are any good. I have dial up connection and don't look at videos just to check them out, takes too long.