No, nonstick cookware contributes no significant fluorocarbons to your diet. Nonstick cookware uses PTFE, which is a polymer that doesn't release its constituent fluorocarbons under any circumstance. Even overheating a pan won't release any significant amounts of PFAS. It will release other poisonous chemicals, but that's a different issue.
The water filter is definitely a good idea. I use a big carbon filter in my kitchen, and that's the only tap that we get drinking and cooking water from.
Then you'll be eating small bits of PTFE (commonly referred to as Teflon), which is completely harmless and will pass through you unaltered.
PTFE is a fluoropolymer. It is a completely different substance than the chemicals we call PFAS. PFAS chemicals are sometimes used when applying PTFE to surfaces, particularly as surfactants, but all current manufacturing processes ensure that no residual PFAS is left behind in PTFE-bearing products. As far as I know, nobody has ever found residual PFAS in cookware, for example. In over 50 years of use in cookware, there is no evidence that PTFE-based nonstick coatings have any negative health effects whatsoever, as long as the cookware isn't overheated. PTFE is among the most stable and immortal of all organic materials, and is logically unable to interact chemically with biological systems. You can eat all the PTFE you want, and it will never show up in your blood stream. PTFE is widely used in medical devices that are implanted inside the body, where it's inertness makes it the ideal coating. It's as safe as any material can be.
Unfortunately, non-stick cookware is pretty easy to overheat, and when that happens it can release some pretty toxic stuff such as fluoroacetic acid. But this has nothing to do with the PFAS chemicals we're talking about here. There are actually very few reports of any real-life health effect from overheated nonstick cookware, and those we do have involve only temporary flu-like symptoms.
The real hazard of PTFE is not in its end use, but rather it's an environmental hazard from its production. It is safe to eat Teflon, but I would never want to live downstream from a production facility.
This is completely different from products that use PFAS chemicals. Prominent examples are outdoor clothing, stain resistant coatings for furniture and carpeting, firefighting foam, and many brands of ski wax. These products contain large amounts of PFAS, which can easily end up in your body. The effects aren't well known yet, but this may well be an global health and environmental catastrophe. But again, that has little to do with PTFE. Two different things in this regard.
