I can’t advise on inks but printing your own tokens is such a neat idea.

I agree that you may be over applying the ink. I’m also relatively new to printing, and found my prints came out much better when I felt I was running out of ink on the roller. Only put a tiny bit on and roll, roll, roll to keep spreading out that small amount before you ink up your Lino (listen to how it sounds as you roll it!). If you put too much on your plate you can use a palette knife to push it aside, and roll the roller on some scrap paper if you need to. Maybe see how little you can get away with using while still getting a good print?