The car began to shimmy and shake while driving.

Of course it is. No car comes from the manufacturer that way. Have it inspected at a tire and alignment shop. They will identify loose and worn parts that are a safety concern long before you hear noises or develop bad tire wear. Any kind of shaking means the steering and suspension systems were ignored too long. They will normally get a quick glance during a routine oil change, and a more thorough inspection will be recommended when necessary.