I am looking into whether there are inner tubes that are fully flat proof. Do you know how much they would affect performance and comfort?
Yes, they're usually solid rubber and they're very heavy, hard to fit and have a lot more rolling resistance than you're used to. Companies like "airless tyres" make them, and I suggest trying to fit them yourself before taking them to a bike shop and paying whatever they ask to have them fit the solid tubes.
You can also get puncture resistant tubes that have thicker walls than normal tubes, and while they are heavier than normal tubes they're nothing like solid tubes. They also don't work very well. You're better off with puncture resistant tyres.
Finally, you can get sealants that you pump into the tube that will leak a little then seal, like Slime. Tubeless tyres use a similar sealant, but swapping to tubeless is a major change and there are many sites describing that process
Assuming that basics are in order and that you are using properly sized tires and inflating them to proper pressure, typical route to get more puncture resiliency is to use different tires with better puncture protection, not different tubes. It reduced my puncture frequency dramatically, I expect it would also help you.
They are also solid one-piece tires made by Tannus (http://www.tannus.com/#tires) and AmeriTyre ( http://www.amerityre.com/our-flat-free-products/bicycle-tires) that will replace the whole tire and inner tube. One thing to consider with these kinda of tires is that the depth of the rim must be meassured properly otherwise the tires will roll-off the rim if it's too loose. They also have a higher rolling resistance and will take some getting used a "smushy" feeling while riding.
Acme Tires (http://www.acmebikeparts.com/) offer flat-free tires and wheel-sets packages if you don't want to do it yourself or take it to an LBS. These are on the expensive side.