If you've ruled out insufficient cable tension, derailleur hanger alignment tools are cheap and accurate. As others have said, start with this task, otherwise you other attempts to adjust tension, limiter screws and cable clamping may be in vain and need repeating if the derailleur is realigned later.
Your hanger might get knocked out of alignment if you drop or lean your bike on the drive side, or it takes a whack at a bike stand, etc. You want to be confident you are starting from a clean slate when troubleshooting these issues, especially if it's not just something you've done wrong after tinkering. If you don't have the tool, borrow or get one! Get a cycling maintenance book and follow the rear mech setup to the letter, ideally with new cables and you'll know that if you still have the problem, it's got to be an issue with the mech itself. It may be worth lubing the moving parts of the derailleur. Salt on roads this time of year can lead to the springs seizing, in which case, heavy dose of lube and manual manipulation might work it out, but in most serious cases, you'll need to replace the mech. This of course, should be your last resort.