Well, If you use the press correctly, there is generally very few if any grounds in your cup - I have left my coffee to stand over 20 minutes, and I don't consider it "spoiled." And if you are only brewing 4 cups at a time, twice a day - that is not much really anyway. I drink that all in one sitting sometimes. So, I wouldn't think there is much chance of it spoiling.
but, it is true that the coffee is stronger. If you don't like that - then perhaps drip is the best option for you. Although a lot of how strong a coffee is has to do with the blend and how long it sits with the grounds.
I'll take French Press over drip any day.
DS