How do I get my painted furniture to not be tacky?

Depending on what you painted it with in the first place, I think you could cover it with chalk paint and then a sealer. I've had better luck with sealers rather than wax on table tops.

Here is an article that might help - https://homeguides.sfgate.com/dry-paint-still-tacky-80720.html

I would not simply paint with chalk paint over the tacky surface that you have now because I think you would have a bigger mess. I think you might need to do some sanding to eliminate the tacky surface of your initial paint job and then use chalk paint. You don't need to sand down to the wood just sand enough to remove the tackiness. Some good prep is going to be the key to sucessfully redoing your paint. You don't mention what finish your dining set had originally and what you used over the original finish and how you did it. Obviously something went wrong somewhere! Perhaps if you give us some more info we can be more helpful!

You’ll need to do a little bit of removal of the old before doing anything new otherwise you may create just more problems. Prime after then paint.

I'm curious what you sealed your painted project with?

You might be able to spray with Shellac. Let dry and when it feels like a smooth finish paint with a good brand of chalk based or mineral based paints. Annie Sloan, Jolie, Dixie Belle are all great lines. :)