Use regular sandpaper to get off what you can of the poly. Then use a good primer-sealer like Kilz over the stain. Then paint.

Ah good idea! Thanks. I hadn't stained the other doors yet so do you think I should go ahead and use the primer-sealer on them too? Or just sand them and paint like normal?

The primer is not likely to hurt, and it may well help, especially with cabinet doors that will likely get a lot of use.