I cannot figure out if the reason the disposal side has a slight odor after a couple hours of being dry is because of the trap setup? Both traps go through the wall conceivably to the a stack. I can check from underneath to see how they are plumbed or is that ok since one trap doesn't feed another? My thought is that the higher trap can create vacuum on the lower trap because they are so close and I don't know if there is a vent or if I need to add an air admittance valve. There is no wall above the sink just an island so there isn't a vent going straight up to the roof for sure but possibly one off to the side (if that would still work?).
AFTERTHOUGHT: If I had to guess the house had a single basin sink at some point and the left drain was the original and the right one is likely looping over and into the left one. Just a guess.