Especially when there aren't enough transactions to fill up a block. So does this mean that, say in the case when transactions in the mempool are low, each time a new transaction enters the mempool the miners will stop attempting to hash the old block and start attempting to hash the new block that includes the new transaction?
The specific transactions that are included in a candidate block are updated far more frequently than once per second. Miners (specifically mining pools) try to maximize their transaction fee revenue, and will adjust their block candidate accordingly.
The reason why blocks are sometimes empty immediately after a block is found, is because miners are blindly accepting that a previously found block is valid, and is mining on top of said block prior to confirming which transactions have been confirmed. Once miners validate a recently found block, they will update their database so that their new block candidates will reflect only valid transactions.