Why did my newly applied paint run and not adhere to the wall?
My guess would be that you didn't prime and that you tried to paint over oil paint with a latex paint or some incompatible combination of paints like that.

You should figure out what type of paint was used for the original coat and purchase a suitable primer to go on top, and then apply your new color.  If you really aren't sure, someone at a paint store can recommend a primer that is suitable for a range of scenarios.

Based on the picture I would probably avoid a paint-primer combination product.

Primer is typically pretty cheap so aside from the additional labor and time, it should not add considerable costs to your project.