How to fix sticky paint
It sounds like someone has painted latex over an oil-based paint. One way to tell is cut out a small bubble, or scrape some of the top coat away (in an inconspicuous place) to reveal the underlying paint. Take a cloth and put some denatured alcohol on it. Rub the underlying paint for a few seconds and look at the cloth. If the cloth has the same color of paint on it, then it is a latex paint. However, if the cloth has no color on it, then the base coat is an oil paint.

If the base coat is oil, you will need to strip off the top coat. It should be really easy to strip since it isn't adhering anyway (it is rolling off when you sand it). I would suggest going back with oil paint. It is highly effective in damp settings and will last for years. It's also really tough and will stand up to cleanings. However, if you prefer latex you will need to do as you suggested and use a primer between the oil paint and the latex top coat.