I use Gina K's 120# weight for all of my card bases. I tried my Copics on it and the ink sat up top of the paper and got shiny too. I use either Neenah classic " Super Smooth" ( not the regular Neenah classic) or I use Hammermill "Color Copy Digital Cover". Both work well for me.
Another thing that can cause the ink to get thick and shiny is when was the last time you cleaned the inside of your marker caps?
If you have not cleaned them in a long time the ink can build up in the caps and prevent a tight seal when you put the caps back on. This allows the air to dry the ink in the cap and the tip
The inside of the marker caps needs to be cleaned every few weeks if you use them a lot.
If you have never cleaned them, Dip a Q-Tip in 91% Isopropyl Alcohol and twirl it around inside the cap to get the old ink out.
Hope this helps
"I have not failed . I've just found 10,000 ways that won't work" --Thomas A. Edison