agreed. While there are things you might do, the risk:reward is not in your favor.

"Not looking to refinish it, maybe just add another coat over whatever is there."

What you have is 30 years of various commercial and home-made polishes and waxes on top of the original finish. It usually doesn't work to apply another coat of anything because it dissolves into the old stuff and makes a gummy mess.

They are veneered, so sanding is tricky.

I would clean it thoroughly - first with water and a mild detergent (oil-free, not that Murphy's crap) and then with mineral spirits to get old dirt and crud off. Then wax it well with good furniture wax and enjoy it.

Use place mats and hot pads to protect it, like any table, and don't let the kids use it for a craft table unless you cover it. A cheap clear shower curtain makes a good protector.

Is there a brand of furniture wax you prefer? And will I need to rent a buffer, or is elbow grease sufficient to prevent swirl marks? A shower curtain is a great idea I had not thought of. Thank you to everyone for your help with this!

This was an interesting article on furniture wax; it gives some brands.

http://www.woodfinishsupply.com/MagWaxArticle.html

Well, that was an interesting article. I read lots of articles and they all seem to have different opinions. A wax layer should only be a few molecules thick so "damaging the wax and not the finish" is just so much hoo-haw. It does lubricate though, so it makes the surface slippery and less likely to scuff. Whatever your selected approach, from nothing to silicone oil, I can find an article that suggests that as the best.

One thing you don't want to use is an old-time recipe of vinegar and linseed oil, sometimes with additional ingredients such as "butter of antimony" whatever that is.

I did some googling on waxing furniture, and those websites point out that it is not the best product to use for a dining table that may have hot or wet items on it...so I will follow klem1 and bobismyuncles advice and just enjoy it for as long as it lasts. Thank you