I'm not personally a fan of the built in soap dispensers, so I just have my dish soap and hand soap on the counter next to each other. It doesn't bother me in the least. My last house we rented had the built in soap dispenser, so I just kept hand soap on the counter next to the sink.
We use the built-in dispenser for hand soap (it's MUCH more frequently used than dish soap during any given day here). We keep dish soap in the cabinet. No clogs, easy to top fill, we're happy with it.
This is exactly what we do. Hand soap in the dish soap dispenser. Bottle of Dawn underneath.