I have a suspicion that you might have to sand deeper than a light surface scouring to get rid of the water marks. My guess is that it is finished with some flavor of boiled linseed oil (BLO). You don't see a lot of tung oil on production furniture because it takes a while to dry. Short of sanding the entire thing to have one cohesive appearance, you might be able to get away with just sanding the top and using Watco Danish Oil in the 'natural' color. Danish Oil is mostly BLO with some varnish additives to harden the finish. Imparts a very mild amber tint as well, which may serve your purposes here. Danish oil is also very easy to apply. You essentially wet a rag with it, or even pour it on the surface, give it 15 minutes or so to absorb, and then wipe the rest off. A couple hours between coats, with sanding not necessary between them. The wood will only absorb as much oil as it can, and because is is not a film finish, you don't need to worry about building or leveling.

Another possibility is Waterlox, which is a combination of tung oil, resin, and drying agents. Waterlox is fairly tough as oil finishes go, though nowhere as tough as a poly finish. It builds a bit, and you do have to level it before the final finish. It also takes longer to dry than Danish Oil, and it is does not like being applied in anything below mildly warm temperatures. But it is pretty tough and water resistant, and is commonly used as a furniture finish for one-off and non-production furniture. For this application, though, it may be too dark.

I love Danish Oil for small projects and things that aren't going to be subject to heavy wear and handling. But any furniture piece with enough flat surface area for a kid or spouse to set a drink on gets several coats of poly. You can buy BLO or Danish Oil at any Home Depot or Lowes. For Waterlox, you may need to go to a specialty store like Woodcraft, or order it online.