This one is actually a pretty traditional/old school jazz box, hollow body, floating pickup, even the knobs are of the style used on 30's and 40's vintage guitars. Take a look at a Gibson Le Grande.
You can get inside tuners and tighten them up. I suggest starting there. Suspect that sometimes people spend money on new tuners when they probably just needed to tighten up the ones they already had. I don't know that adjusting the truss rod would affect how long the guitar stays tuned unless the screw is completely loose. But if the neck mounting screws are loose that could certainly affect tuning stability as the neck pivots a little in the neck pocket. Might try snugging them up a bit if the neck looks like it has shifted at all.
You can get inside tuners and tighten them up. I suggest starting there. Suspect that sometimes people spend money on new tuners when they probably just needed to tighten up the ones they already had. I don't know that adjusting the truss rod would affect how long the guitar stays tuned unless the screw is completely loose. But if the neck mounting screws are loose that could certainly affect tuning stability as the neck pivots a little in the neck pocket. Might try snugging them up a bit if the neck looks like it has shifted at all.