Yikes!
I'll tell this as an extreme example. When I first bought my Jeep Wrangler it was 12 years old but low mileage one owner. The tires looked very good so I didn't think anything of it. Turns out they were original. I began feeling a slight vibration that was getting worse. Thinking it was a frond-end issue (common on Wrangler TJs) I made an appointment to have it checked out. On the way to the shop driving on the turnpike, the left front tire threw its tread! It ripped off the front fender and broke the step bar in half. I was able to keep control and get to the side of the road without hitting anyone but it could have gone much darker.
I've also had older tires on cars in the past that caused vibrations and ended up with either separating belts or bubbles in them. (admittedly doesn't sound like your vibration problem, but still....)
If those are indeed original tires I wouldn't drive another day without replacing them, even if you have to go cheap (I did on my daily, not wanting to continue on the aging Michelins and the cheapie tires have been just fine).
