Hi Emery and welcome to Ninjette!
You've got multiple issues, so let's address them individually:
1. spark plugs and gap - Pull plugs and measure their gaps. Post picture of their condition. Make sure they are not
Champion spark plugs, which can easily destroy your engine if you use them.
2. petcock - should only flow when engine's running. Disconnect hoses from carbs and test by applying vacuum on smaller hose on petcock to start flow from big hose. Remove vacuum, does fuel-hose stop flowing? Might be worth disassembling and seeing if there's some grit on sealing surface keeping plunger from stopping fuel-flow. Also stretching diaphragm spring sometimes help it seal better. If these efforts aren't sufficient, you can get petcock rebuild kit with new seals and filters will restore 100% functionality.
https://www.ebay.com/itm/323531028088
3. new slide diaphragms. Luckily you can replace just rubber part.
https://www.ebay.com/itm/332406608600 Trick to replacing is soak in 180F hot-water for about 5-minutes. Then stretch diaphragm sideways so inner opening is long slit. This allows you to slide it off and on centre part of slide. I recommend making holder that grips slide in vise to make diaphragm replacement easier. I didn't get pictures of this procedure, will do next time I perform it.
4. carb-rebuild. Float-valves aren't sealing off fuel-flow. Most likely need float-valves replaced and float-bowl o-rings as well along with those pilot-screw O-rings you ordered. Very likely that jets and fuel-circuits are clogged as well. Note that "carb cleaners" don't actually do anything nowadays after removal of chlorinated compounds. You'll need ultrasonic soaking with much more reactive solvents along with mechanical flossing with guitar wire. Complete refurb job will restore factory-fresh running condition like when it left showroom floor.
Search for "clean carbs ducatiman" on this forum to find similar issues as yours and their simple resolution.