DueFlashStorage will work but you do need to keep in mind a couple of things
1. Yes, there are limited writes to FLASH. I think you can program it about 10,000 times. So, every second would be a bad idea for sure. EEPROM is longer lived, an external EEPROM module should be able to handle 1 million writes.
2. In order to upload a new sketch you must erase the entire FLASH memory first. You'll lose all of your stored data.
Your best bet is to add an I2C connected EEPROM chip if you can. They last longer and they don't lose their settings when you upload firmware. Even so, 1 million seconds is 11.57 days so if you're writing every second and you leave the thing on you'll ruin the EEPROM within two weeks. As such, it is better to either only write when something changes or try to write much less frequently. If the board could lose power and you want to save changes you can rig up a circuit that detects that the power is gone and tells the processor to quickly drop everything and write to EEPROM. This allows you to only write when the power dies rather than all of the time. Otherwise the settings and values can all persist in RAM.