Having to hit the breaks hard to slow the car down to stop the bouncing to me sounds like too much of a severe problem than just worn out shocks.
I would get your entire front suspension checked out to see if something isn't broken, become disconnected, etc.
Also, worn out tie rod ends can also cause relatively bad tire wobble. It feels like your tire is simply out of balance, but in reality, with a worn out tie rod end, you tire will wobble from left to right instead of bouncing up and down.
2001 E430, Bourdeaux Red, Oyster interior.
1973 280SE 4.5, 170,000 miles. 568 Signal Red, Black MB Tex. "The Red Baron".