Not saying you do not have yet another "event" to use as a reminder at some point your hubby least expects it but if the paint is coming off down to the surface you have a major adhesion problem he did nothing to cause. So blame him only for the drips, sags and all.
You cannot fix surface adhesion by adding paint layers over the top and expecting some glue like magic to happen so the answer to your question is you should take the paint off down to the laminate and start over.
I am not sure the Zinsser primer, eggshell latex and acrylic, and your plan to coat it after it has cured for 30 days with poly is the right product mix for this. I do understand the paint was free but it is probably not going to work.
You could try a higher bonding primer but if I were you I would shift to a paint or primer for plastics for this project. Melamine will work but it is expensive, hard to work with, and chips easily. You can get it mixed to any color though.
Rustoleum makes paint and primer for plastics. I think the paint comes in rattle cans only but you can get the plastic primer in either spray cans or liquid form. There are a limited number of colors. You can paint over the primer with anything.
Krylon Fusion is great stuff and comes in many colors (see chart but I am not sure it is the most current) but in rattle cans only. It was developed for painting plastics and laminates but you can use it for other things. I have had great luck with it even restoring outdoor plastic play sets and furniture. It works great on laminates and I think would be an excellent choice for your project. I believe they make a clear coat if you wanted to add one or change the sheen. Most colors will come in gloss only.
You will need a few cans and with any rattle can products you want to apply many thin, even coats and not overspray at once or the product will drip. Spray paints have quick recoat times so you can accomplish painting your dresser easily in a day.