I am working on my first paper model kit (Princeton Brewery). I dutifully went out and bought a grey marker to color the opening and cut edges so the white paper doesn't show. The problem is, though, that the paper absorbs the ink, and you can see it darken under the printed detail on the front of the cardstock.

The marker is from Letraset.

I have tried doing it as quickly as I can, but still have the problem. Any tips?

I've found that colored pencils work well to "cover" the edges. Grey, brown, tan, etc., depending on color of the part.

Hi Jeffrey
Though I'm not familiar with the pens you mentioned, I'm very confident that I know something about them. I'm willing to wager that they are alcohol based. While usable, alcohol based pens flow so much more easily than water based, they do exactly as you described. I'm also guessing that the Gray color you have is much too dark. A VERY light gray is what you want. A good choice is something like a "20% Warm Gray". You can go as high as 30%. Here's something that you can experiment with. Don't try to apply the pen to the edge of the card. Do it on the back and let the color seep through. This has worked for me in the past, but it does take some learning. As mentioned in the previous comment, pencils do work well too, so does paint or chalk. For my money, try water based markers. My personal preference are made by UCHIDA and are their MARVEY LePlume II brand. They are available from <bestpensonline.com>. About $4.00 each.
Good luck.

I looked up the marker on line, and indeed it is alochol-based as you suspected, Dave. I'll try some of these other suggestions. The grey itself is not that dark.

Chalks can be blended and work well too (But can also be a bit messy.)

I have been using acrylic craft paints for year. I take a small pointed brush, dip it in water, lightly dry it, dip it into a glob of paint and get most of it off the brush, then, applying from the back of the piece, run the side (not the tip) of the brush over the edge. I have had good luck with the method. If you have seen my tutorial on the J&RG facebook page, you can see an example. At any rate, it's another alternative. My favorite colors are mudstone, neutral gray, light gray and charcoal, depending upon the piece being painted.