Unless I misunderstand you or my figures are way off, I show that a 2 x 8 on 24 centers will only span officially 6 feet, and you are saying they are spaning 11ft 4 in????
Ok, enough about the spaning, the problem is already created and you want some means to strenghten the drill hole area.
The term is called, "sistering" the same as your term sandwiching.
As we both agree, the sturdyness will improve by "sistering" from one end to another, but that is alot of time, work and WEIGHT.
What I would do is to get some 3/4 or 1 inch plyboard and go three feet on both sides of the drill area and on both sides of the 2x8. Then I would screw 2 1/2 inch heavy duty deck screws into the plywood through the 2x8. Then I would screw one screw per one inch square area. That is many screws but this might be what you need.
I would also consider not putting heavy furniture on that side of the wall area above the drill holes. I am talking about a piano or billards table or the like.
Unless I am very off base and mis understanding you, that span is do wide, the bounce and play will always be there.