I would refine the info you have been given to say tire balance gets sensitive only if you exceed factory air pressure specs especially on the front 29psi. That being said I always get the tire shop to "balance" my bare wheels first to see how far from perfect the wheels are before mounting a tire on it. (do not add balance weights until after tire is mounted then balance wheel & tire assembly)
I would check/replace the front engine mount if tire pressures are correct. If you do a search for motor mount you will see all the strange things that go on when the mount wears & its not to hard to change.
