How much wine do you have? If it's a lot, I'd simply decant wines before drinking, leaving sediment behind.
If your wine is from this fall, you're not bulk aging long enough. I was originally taught to never bottle before 4 months. I've tried various periods, and came to the conclusion that even with modern fining agents, 4 months is a minimum, and red wines will have 6 to 12 months before bottling. This provides sufficient time for sediment to drop.
A test for tartrate crystals -- eat a few -- they'll crunch like tasteless sand in your mouth. A 3 yo wine dropped a half a dozen crystals in the bottle and I got to experience this last week, so the experience is fresh in my mind. Alternately you can do a visual inspection, see white flakes, and simply trust that it's tartrate crystals. [Go with Plan B as you won't have to rinse your mouth out.
As to why the oaked wine dropped crystals and the unoaked didn't? Chemistry. Something in the oak produced a nucleation point, so the crystals formed. My guess is the unoaked wine will form crystals at some point, since it's the same wine, so the acid levels are the same.
NC Muscadine is high acid, and my guess is that TX Muscadine is as well. Cold stabilization is the easiest acid reduction method and has the fewest side effects -- chill the wine to to the 30's F for a week or two -- the drop in temperature will lower the acid saturation point and crystals will drop. Rack off the crystals.
When I lived in Upstate NY, I put carboys of whites on my porch for a week or two, as my porch was in the 30's during the winter. My guess is that TX doesn't get that cold, so you can put the wine in 1 US gallon / 4 liter jugs and refrigerate. That will do the same thing, although if you have 50 gallons of wine it may not be the ideal solution. Many moons ago an acquaintance purchased an extra refrigerator, took out the shelves, and it fit two 19 liter carboys.
Note -- although "full" cold stabilization works if the temperature is close to freezing, it's not essential. If you can get the temperature below 50 F for a couple of weeks, it's likely your wine will drop enough crystals that you won't have any form in the bottle.
My cellar is typically 58 F in the winter, and over years wines with relatively low amounts of acid may drop crystals.