Why would a bookcase I painted months ago still feel tacky?

That seems really odd but a polycrylic sealant over top should take care of it.

Depends what the bookcase is made of. Solid wood should not be a problem. Fake woodgrain over particle board or MDF would be. The fake wood should have been primed then painted. The all in one paint won't work. What you can do is wipe it down with a lint free cloth and Goo Gone to remove the stickiness. Let it fully dry. Then two or three coats of a water based poly should do the trick.

Have you tried a hair dryer on it? If you live in a humid summer area it may be that the paint just isn't dry yet or if you put on more than one coat the first coat was not dry enough before you added another coat. If you added a third coat in rapid succession that may also mean that all the layers of paint have not dried yet. A hair dryer may help. It may have been better to have used a stand alone primer first rather than a paint with a primer in it. Paint with primer is really just a thicker version of regular paint. It doesn't work the same as a separate primer and then paint. I find that if you are repainting furniture over old paint--even if you have sanded--it is better to use a separate primer before using regular paint.

I would paint it with 1-2 coats of polycrylic to seal it.