Why is my new toilet running so much?
Judging by the amount of scum/sediment on the bottom of that tank, I'd bet just cleaning the mating surfaces between the flapper and the outflow drain will solve the problem. And if it only took a few months to get that scummy, you'll want to plan on cleaning these parts regularly—this is especially prudent if you're on septic.

Side note: You might want to look into why your water has so much sediment in suspension. Are you on well water, by any chance?