GF recommends that you replace the doors if you want to use a white paint like General Finishes Antique White Milk Paint. Some things are not meant to be and painting charred doors is a recipe for trouble. The charring will bleed through.

"Cure time" is the amount of time needed for the finish on a piece to reach maximum hardness and be ready for normal use. This is different than "dry time." 

Water-based finishes cure faster than oil-based finishes, approximately 21 days vs 30 days under ideal conditions (70 degrees and 70% humidity).

During the curing stage, treat your project with special care. Avoid placing heavy objects on it. Always avoid the use of furniture polishes, dusting sprays or oil soaps as they may contaminate the finish for future re-coating. Use a water-dampened cloth for regular cleaning. Wipe up spills in a timely manner.

WATER SPILLS AND DAILY USE: Clean water spills up as they occur during regular use and protect from heat. Although General Finishes products are water resistant, they are not impervious to water. Surfaces will not be damaged by spilled water so long as it is cleaned up promptly. Use coasters to protect your wood from standing water and pot holders to protect from heat