Darting from side to side would not be swaying. You problem sounds like either the toe setting is off a lot or something may be loose in the front. Did the problem just start happening ? Any recent work done ? Tires are the same brand and size on each axle ? What size tires are they ? The factory size will be on the driver door jamb. Should be F78-14/F-70-14 on V-8 cars. The modern size would be 205/75-14 / 215/70-14. Tire are less than 5 years old ? Air pressure is adjusted correctly? All speeds ? Does it all the time not just when changing lanes ? Is the car at stock height? Odd that you said that the ball joints are tight. Has the front suspension been thoroughly inspected ? By a front end mechanic ? Have access to a lift ? Alignment done lately ?
I agree with Moparleo, there is a few different things that need to be checked, make sure all the front end parts are good, tire pressure all even and are not cheap off brand tires, make sure your shocks are good and the alignment controls how the vehicle drives, if the toe is not set correctly the car will fight you going down the road.
I had to stock fiberglass 15's Goodyear tires on my 70 cuda and chase driving that car for years got tired of it and put on Mickey Thompson sportsman radials. It's a dream driving now and better traction.
A simple check is to Jack up one side or center and shake the tires for movement.