The Waterpik is the most nature friendly method of flossing over string floss. The Waterpik is also 50% more effective in removing floss than string floss. There is a model of the Waterpik that flosses and brushes at the same time.
To use tooth picks from certified wood, I would say! Maybe that there is some natural floos like hennep strings, should be good to I think
Don’t worry about the environmental impact of your flossing. That is likely the least impactful thing you do. This is coming from an environmental scientist.
A water flosser is a really good option but expensive. If you can't afford that a lot of eco stores have plant based floss or biodegradable floss.