You should use a Primer-Sealer. If you told the Paint Retailer what you were painting, he should have given you the right Primer-Sealer. Re-check the can. The paint should be stirred. I don't think anyone can 'shake' a can of paint like one of those can-shaking machines. Depending on how much paint is in the can, by now, the 'mix' may be off. In other words, in shaking the can during the PREVIOUS 2 coats, you might be 'left with' paint ingredients that aren't, well, paint. You know how paint settles-out when it sits for a while??? The paint may be a bad mix now ... The veneer can be anything. I could be plastic or it could be wood. I could have had polyurethane on it, maybe not.
Since I'm gonna assume you went to a Sherwin-Williams Retailer, I'm gonna assume the problem is not with the primer or the 'type' of paint. I'm gonna be the problem is with what's left in paint can. (A bad mix.) Humidity / moisture can cause paint to be tacky. Water is a component of latex paint. That's why I'm guessing the shaking of the can, over time, didn't mix the paint very well. (You could be left with a lot of water and little paint in the can.)
You may need to get another can of paint, and do stir it. (The can has directions on it describing how to use it. Sounds silly, but as you can see, not following them can cause a problem ...) You may get away with just putting another coat of paint on it but then again, you may have to put on a coat of Primer-sealer first. Check with the Retailer. If you can, bring in a drawer to show him the problem. ANd bring your old paint and primer in with you too. He may just 'give' you a free can.
My best to ya and hope this helps. DO wait for others to Post a Follow-up. They may have more insight. (Give it a day or so ...)