Twitchy handling Sedan / carsick kids, headaches

[theraymondguy]

2007 Sedan, auto with power group. Factory 15" alloys, stock tire size and recommended inflation. Generally it has no weight in trunk and is driver only occupancy. Car is box stock with the exception of synthetic engine oil.

It has been an overall fantastic car. We're the second owner, have had it for 2 years.

The car is not driven in a spirited fashion, though I'd like to use it for our "Florida Vacation road trip car" which has 2 opportunities:

My daughters (9 and 12) complain of headaches when road tripping, though no complaints when travelling in our '05 Kia Sportage. The Sportage has rear tint, I'm thinking this may be all that is needed, but was wondering if others had the same issue.

I've noted the car feels twitchy at highway speeds when road tripping, like it could change lanes merely at the suggestion. Playing with tire pressure hasn't had much effect, tires are Hercules brand non-directional. Rear suspension feels soft (has always felt soft), perhaps more so as our girls grow.

With a modest budget, what would be the most likely fix? Swaybar? I could swing a tire swap (Always happy to install Michelins)? I don't want to get heavy into suspension mods or lowering (My wife likes the stock ride height).

Thanks for input!