There is no correct answer. I will give you my opinion, which is just my opinion. There will be (or should be) others that might have a slightly different take. The issue about wood warping if there is a different finish on both sides is quite correct, especially if there is little in the way of physical restraint or the design does not account for it. Slow drying finishes will smell for a while, and the smell never seems to go away with some finishes. Clothing and other articles may pick up odors or even become soiled by weeping finishes - linseed oil is notorious for that. Extreme temperature and humidity changes bring out the worst in wood and finishes. Finish choices are to some extent dependant on what you are going to use the cabinet for: Clothing, dishes, children’s toys?
Here is my advice. You can finish the inside of the cabinets with shellac, which is fairly benign stuff and will dry without an odor, though it can be damaged by alcohol. Shellac, especially if it is from an old can, can get gummy in high humidity and heat, so be careful. It will have a different ability to slow moisture exchange depending on the outside finish, but design considerations will help prevent warping – frame and panels, physical restraint and so on. I would not finish the inside or outside of the drawers with anything, except for the front. The front is usually stable and restrained by the sides, depending on the design and size, and it is not likely to warp from moisture variations related to the difference in the finishes. I’ve never had one do that. Recall too that some woods, like cedar, are used specifically because they do bring a pleasant odor to the contents of the drawer.
As you drew it, the top appears to have a real potential for warping, and I’d make sure you finish both sides, and using the same substance would be ideal. The panels in the frames are probably going to be stable since they appear to be small and adequately trapped by the frame.
Given the myriad of finishes available, only experience can tell what works the best.
Finally, the design looks like it has the grain of the drawers going vertically. If that’s the case, and not just a rendering issue, avoid doing it. They need to be horizontal and match the drawer sides, otherwise there will be differential wood movement which will be worse than any finish induced problems.