I fear your spongy floor is due from the 2x8 joists themselves. A 2x8 joist is 7 1/4 inches deep, and if these are placed (I'm assuming)16 inches on center over a 12 foot span, the deflection under load will be marginal at best. Your idea of 1/2 inch osb would help, but the increase in stiffness (approximately 1/3 stiffer) would probably be disappointing to you. You might try installing the bridging first, as any missing bridging will make a tremendous difference. If that isn't enough improvement, An additional solution would be to glue and screw 2x4's to the bottom edge of the joists. The 2x4's would be the flat way, so the joists at the bottom would look like an "I" beam. The 2x4's should run the entire length of the joist. This will reduce your head room by 1.5 inches, but would make a significant increase in stiffness over just the osb solution.