Dredging would only work if, in your example, the bed of the river was lowered by 5 feet all the way to the sea - have you thought about the cost compared to the actual damage caused by flooding.

Estimates are that flooding causes just over £2 billion of damage each year - currently around £800 million is spent each year on flood defences - this is mainly spent on densely populated areas making the spend cost-effective.

Just like coastal erosion, judgements have to be made on whether or not spending public money is worthwhile.  For example tens of millions have been spent on coastal defences around Brighton whereas just a few miles up the coast houses have been allowed to fall into the sea.