Some photos would be helpful as "old worn look" is not specifically descriptive. Do you know what type of wood it is? I'd hold off on the steel wool. Some woods (for example oaks, that contain a lot of tannin), will react with the iron in the steel wool and leave black spots.

First, your finish will never be better than your preparation. Just cleaning and sanding will not effectively remove the old finish, in general. And a finish will not smooth out a splintery surface.

Second, be aware that most products called "Tung Oil Finish" are not tung oil, and most do not even contain any tung oil. Most are either a thinned varnish or a linseed oil-varnish blend. Unless it says 100% tung oil and contains no petroleum distillates, you are getting an imposter. Something that "looks sort of like tung oil." Tung oil, the real stuff, is not very protective and if you don't apply it right and allow sufficient inter-coat dry time, it will not look good and need to be stripped off.

Varathane is a brand name, not a specific product. It's sort of like saying, should I buy a Chevy? Well are you talking about a Volt, a Corvette, or a cargo van? Varathane makes both urethane and acrylic (water-borne) varnishes, as well as stains and other things.

It would also be helpful to know the look and feel that you are aiming for. Since we don't know that the current appearance is, it's hard to say what is going to change it.

I can't add anything Bobsmyuncle hasn't said, because I'm still learning.
I can tell you I've found two methods to get a completely matte, sealed finish, but you've got to do your work first.

Where are you on this project? I'd love to see pictures.

Oh - I've also found that before I go hacking finishes with sanders, steel wool and other aggressive things, that something as simple as washing it down with Murphy's Oil Soap will cause centuries of grime to just wash off, without damaging the wood.

I picked up an antique, two sided/two person desk (drawers on either side) with gorgeous rounded corners and beading applied. I fully intended to paint it until the gorgeous, amazing veneer and finish were revealed.

Just a thought.