Why does my paint bleed past the leading lines on faux stained glass?

Yes, the paper is most of your problem. Regular white cardstock is porous, so any translucent medium you lay down wicks into the fibers and travels sideways under your lines. Glossy or coated stock mostly fixes that because the color sits on the surface instead of soaking in. It can bead up a bit at first, but if you work in small sections and nudge it around with the brush tip, it behaves fine. Vellum is another good option, same idea, everything stays on top and dries as a little jewel-toned pool, which is exactly the stained glass look you're after.

The paper swap is the big one, but the thing that actually saved my lines was switching to fabric paint for the outlines instead of a pen. I was using a fine liner too and mine bled every time, no matter how long I waited. Then I tried squeezable 3D fabric paint in black, laid the lines down a bit thicker, and let them cure overnight before adding any color. The raised line acts like a little dam so the color pools inside each section instead of sneaking under. Takes some practice to keep the line width even, but the crisp edges are night and day.

Not sold on the overnight cure part of that fabric paint suggestion. In my experience fabric paint on paper takes way longer than a night to fully cure, the paper holds moisture in and the lines stay slightly tacky for days. I rushed mine once and the color grabbed at the edges of every line and pulled little chips off. Giving them two or three days made an actual difference. So sure, try the raised line idea, but don't trust the timing on it. Also worth testing on scrap first, since some of those paints stay a bit rubbery on cardstock forever and never feel right under a finished card.

Coming at this from a slightly diferent angle since I've only done the faux glass thing on clear page protectors taped over a template, so I can't speak to cardstock at all. But one thing that helped me there was tilting the piece a little as each section dries, so the color settles against the line instead of pulling away from it. Does that trick work on paper too, or does the wetness warp it? Also, when a color does cross a line on you, is there any way to clean it up afterward, or do you just start the whole piece over? I've been scrapping mine entirely and it feels wasteful.

Update: grabbed some glossy cardstock and tried the fabric paint outline trick from the second reply. Lines are way crisper now, but the person who said the paint stays tacky for days was right, mine still feels soft after two. So my question is, can you heat set it gently with an embossing tool, or does that just bubble the lines? Also tried the tilt trick while drying, that one works great.