Stop using putty. Use pre-mixed drywall compound. It dries faster and is easier to sand. Also, what a lot of people who are novices to filling screw holes is to fill the screw hole with just enough putty and then smooth it with the rest of the surrounding wood. As it cures, it shrinks, meaning you have to fill it again. Instead, fill the screw hole and leave it somewhat mounded. It'll also take a while to cure, which is why I say use drywall compound. It'll cure in far less time, meaning if it shrinks too much, you can easily fill it with more and have the whole thing done quite quickly. Use a sander to even everything out with the sides.
Everything OTHER than screws has some alignment issues or strength issues or a combination of both. Add in not being familiar with a lot of woodworking practices and you've got quite the quandry there. A nice sliding dovetail would give you the nicest looking strong joint, for instance, but is impractical.
What you could do instead of putty is to buy either a plug cutter, pre-made plugs, or a dowel to jam in the holes and saw them flush with the sides. Again, these still require sanding, so I'd suggest skipping those methods, since the only thing they'll end up doing to your cab is making it cost more