what would make my '06 accord feel like it driving down road sideways?

Suspension and alignment are not related to something being the matter with the tires themselves. Try taking it to another trusted tire dealer and see what they say. Maybe also get someone who knows cars travel on open highway in another and CAREFULLY look at your car as you drive at highway speeds to see if they see anything unusual. Maybe even have someone video record you driving to show and record your concerns and show tire guys.

You may have a collapsed brake hose or a caliper piston not releasing. After driving for about 10 minutes, VERY CAREFULLY touch the rims to see if any are hotter than the rest. When you're having it inspected are they actually checking all of your front end parts? Ball joints, tierods, control arm bushings, etc? When I was at the Honda dealer the control arm bushings were a very common issue.

To add to break scenario. Sometimes if brake is sticking you will smell a peculiar order by tire and area affected and sometimes a lil smoke.

Also depends on the road you are driving on. Many concrete roads have grooves in them. Your tires will tend to wander on these grooves. Why do they make them of concrete? It lasts longer, but not as forgiving as asphalt. What kind of road do you drive on? Black asphalt, or concrete?

Thank you all for the good advise. I will let you know how it turns out.