Okay, I just had some success!
1) I ran my folder through once horizontally and once vertically. No idea if that had an impact, but I am mentioning it.
2) I tried my colorbox chalk ink, some Gina K dye ink, and the same Linen Avery Elle pigment ink I tried Saturday. They all worked! The dye ink is just a little streaky because of how it slighly beads up. The pigment ink looks the best, but they all look good. I also tried distress ink and got bad results.
3) I used the same embossing folder I used Saturday!
4) The difference!!??? The paper I used was some in my stash that is not the Neenah paper that I usually stamp on. It is probably some cheaper paper I got at the craft store, it has a slightly rougher texture and it looks like the stuff I sometimes have trouble stamping on because it wicks the ink out into the fibers. So the paper, I hypothesize, soakes the ink in faster! Maybe the Neenah stuff is coated to make it smooth so that does not happen?
5) Other notes: My cuttlebug definitely needed a 1 cardstock shim. It made a difference.
Brayering leaves a thinner coat of ink than pressing the inkpad directly to the folder. Brayering was more even and was less likely go get in the embossed portions. However, there was just a little less coverage on the ones where I used the brayer, but it still looks good.
I am SUPER happy with the results! And super happy I found a use for the cheap cardstock that I won't stamp on!