35 psi?! WAY too much air pressure! Amazing they haven't popped off the rim.
The inscription on the sidewall is the 'max' pressure for the 'max' load rating for that particular tire (What ... 'bout 1200 lbs?). It's not the "recommended" tire pressure; your car must be darting all over the road like it has a mind of it's own ...
I have 205/60-15's, running 24psi front/rear, and that's assuming a 50/50 weight bias, with my fat-ass in the cockpit, at somewhere near factory-posted weight (give-or-take a bucket of KFC). And my current Dunlops are quieter than my previous 195/60-15 Michies. Good wear pattern, with no squirrelly tire antics.
Re-bush the swaybar (if that's in-fact where the rattle is) and set to a less stratospheric tire pressure first, before going to different springdingies. If you ditch the swaybar, you may not like the way it handles afterward.