Yep, from that picture I can see the finish deterioration and I'd strip them. Acetone is what I use. Find the smallest container that will hold them, I've used the plastic tubs that lunchmeat comes in and use the lid to keep unwanted things out. I generally work outside for this stage. Soak your grips for a couple hours then scrub them with an old toothbrush. It may take a couple of soaks but you don't have a heavy poly finish to cut thru so I doubt stripping will be a big job. I'd touch up around the checkering with some 220 sandpaper lightly to smooth imperfections.
Pure Tung oil is kinda hard to find. Tung oil finish is available but I've been happy with an old homebrew oil finish. 1 part boiled linseed oil, 1 part mineral spirits and 1 part spar or marine varnish. Brownells sells a small container of LinSpeed oil finish that is about the same as my homebrew. My mix will work without the varnish, just won't be as glossy.
The way I do it is start with clean dry stripped grips and I usually just use my finger and rub it in. Maybe I'll use a clean lintfree cloth the first application but spend some time and rub it in good and as it soaks in, continue to add more. The checkering is tricky. You don't want to build up too much in there so use a toothbrush and get it saturated but don't leave any globs.
Let them dry for a day then use 500 grit sandpaper or 0000 steel wool to scuff the finish, wipe it down good with tack cloth and repeat the procedure. The reason for scuffing is so the next coat will get into the wood and not just lay on top of the oil. You'll just have to get a feel for how to handle the checkering. Maybe the first coat is all you'll want to put on it. About 3 coats will do for the smooth wood but more won't hurt.
After all is done I wax mine. I use Johnson's paste wax about 3 coats and then a couple coats of Renaissance. I think the Ren makes it pop a little more but it may be my imagination.
Good luck and don't burn the house down.