To address a few questions. Any paint will be sticky if you apply it too thick or recoat too soon. Oil based paint is awesome but very hard to work with. If you are a novice water borne acrylic enamel (I like breakthrough by ppg but there are some SW and benajamin Moore versions pros like more) is a nice mix of easy to work with an durable. They dry fast though so you have to move quickly to keep a wet edge and sand between coats. If you can get one I’d try to use a sprayer. We did to our built ins. Removed everything we could then masked and isolated the rest. That will give you a good finish. We did not sand between coats using the sprayer as the paint hadn’t totally cured and we weren’t trying to smooth out roller stipple or brush strokes.

Take your time. It’s not that hard but get it really really clean and scuffed up very well. And use a good primer. Good luck!