A lot of good info so far, but I can add a little.
When I first got my Dart it would darn near change lanes if I hit one of those reflectors in the road, and the feel of the car overall was exactly like you described.
Scary huh?
My problems were caused by badly worn lower control arm bushings.
Once I replaced those and aligned it with more caster it smoothed right out and got WAY more stable.
I commonly run 80-90 on the more open freeways now with total confidence.
Some already mentioned steering and suspension parts, but since I went through the exact same thing you described, I thought I would tell you.