Greetings & Salutations!!!
So there's some maintenance that needs to be brought up to date, no big deal, all used bikes have them. Take care of worse issues one at time and you'll have perfectly running bike in no time!
Don't use fuel-stabilizer, it's for storing bikes over winter, too late for that as petrol is already bad. What you need is fuel-system cleaner with healthy dose of PEA. One of these should work:
- Red Line SI-1
- Techron Concentrate Plus
- Gumout All-in-One
- 3M Max Strength Fuel System Cleaner #08814
- Royal Purple Max Atomizer 18000
If this doesn't work, then yeah, pulling carbs for full-restoration job would be next